<!DOCTYPE html><html lang="en">
      
      
        
    
    
    
    

      
      
      <head>
        <meta charset="utf-8">
        <meta name="format-detection" content="telephone=no">  
        
        <title>Lecture 8: Graph Theory II: Minimum Spanning Trees | Video Lectures | Mathematics for Computer Science | Electrical Engineering and Computer Science | MIT OpenCourseWare</title>
    <!-- Begin Automatic Metadata Insertion -->
    <meta content="6-042j-mathematics-for-computer-science-fall-2010" name="WT.cg_n">
    <meta content="Lecture 8: Graph Theory II: Minimum Spanning Trees" name="WT.cg_s">
    <meta content="Explores the various measures of connectivity of graphs and how these can be used to categorize and analyze graphs." name="Description">
    <meta content="Leighton, Tom" name="Author">
    <meta content="Dijk, Marten van " name="Author">
    <meta content="6.042J,6.042,18.062J,18.062,walks,paths,connectivity,cycles,closed walk,acyclic,tree,spanning trees,minimum weight spanning trees,Computer Science,Probability and Statistics,Applied Mathematics,Discrete Mathematics" name="keywords">
    <meta content="6.042J Mathematics for Computer Science | Lecture 8: Graph Theory II: Minimum Spanning Trees" name="Search_Display">
    <meta content="Computer Science" itemprop="about">
    <meta content="Probability and Statistics" itemprop="about">
    <meta content="Applied Mathematics" itemprop="about">
    <meta content="Discrete Mathematics" itemprop="about">
    <!-- End Automatic Metadata Insertion -->

	<link title="default" rel="stylesheet" type="text/css" href="../../../common/styles/grid.css">
<link title="default" rel="stylesheet" type="text/css" href="../../../common/styles/base.css">
<link title="default" rel="stylesheet" type="text/css" href="../../../common/styles/menu.css">
<link title="default" rel="stylesheet" type="text/css" href="../../../common/styles/jquery.bubblepopup.css">
<link title="default" rel="stylesheet" type="text/css" href="../../../common/styles/search.css">
<link title="default" rel="stylesheet" type="text/css" href="../../../common/styles/courses.css">
<link title="default" rel="stylesheet" type="text/css" href="../../../common/styles/courses_new.css">
<link title="default" rel="stylesheet" type="text/css" href="../../../common/styles/jquery.jscrollpane.css">
<link title="default" rel="stylesheet" type="text/css" href="../../../common/styles/media_tabs.css">
	<link href="../../../common/xml/ocwcc.rdf" type="application/rdf+xml" rel="metadata">
	<link rel="canonical" href="https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-8-graph-theory-ii-minimum-spanning-trees/">
	<link rel="apple-touch-icon" href="../../../common/images/apple-touch-icon.png">
	
	
      
	
	<script type="text/javascript" src="../../../common/scripts/jquery.js"></script>
	<script type="text/javascript" src="../../../common/scripts/ocw-media-utils-offline.js"></script>
	<script type="text/javascript" src="../../../common/scripts/ocw-offline.js"></script>
	<script type="text/javascript" src="../../../common/scripts/jquery.bubblepopup.min.js"></script>
	<script type="text/javascript" src="../../../common/scripts/jquery-ui.min.js"></script>
	<script type="text/javascript" src="../../../common/scripts/jquery.jscrollpane.min.js"></script>
	<script type="text/javascript" src="../../../common/scripts/expandy.js"></script>
	<script type="text/javascript" src="../../../common/scripts/bubble-popup-offline.js"></script>
	
	
	
    <script type="text/javascript">
      $(document).ready(function() {
        $("#tabs").tabs();
        IpadScroller();
      });
    </script>
    
    
    
    
    
    
      
       
		 

        
        
        

        
        
        
        
        
        
        
        
        
      </head>
    <body itemscope itemtype="http://schema.org/WebPage">
        
	

        <header id="top">
			<div id="grid">
				
				
					
<div id="portletwrapper-6f63772e746f70706f72746c65746d616e616765720a636f6e746578740a2f506c6f6e650a736974652d686561646572" class="portletWrapper kssattr-portlethash-6f63772e746f70706f72746c65746d616e616765720a636f6e746578740a2f506c6f6e650a736974652d686561646572">
<div class="portletStaticText portlet-static-site-header">
<!--googleoff: index-->
<div class="grid_6 alpha" id="banner"><a href="https://ocw.mit.edu/"><img src="../../../common/images/ocw_mast.png" class="logo" alt="MIT OpenCourseWare, Massachusetts Institute of Technology"></a></div>
<div class="grid_6 omega" id="subscribe">
<aside class="module" aria-label="Connect with OCW">
<table class="social">
    <tbody>
        <tr>
            <td class="socialbutton"><a aria-label="Subscribe to the OCW Newsletter" href="https://ocw.mit.edu/subscribe/index.htm?utm_source=header"><img src="../../../common/images/trans.gif" alt="An icon depicting an envelope.">Subscribe to the OCW Newsletter</a></td>
            <td>
<a aria-label="Facebook" href="https://facebook.com/mitocw"><img src="../../../common/images/icon_fb.png" alt="Click to visit our Facebook page."></a>  <a aria-label="Instagram" href="https://www.instagram.com/mitocw/"><img src="https://ocw.mit.edu/images/icon_ig.png" alt="Click to visit our Instagram page."></a> <a aria-label="Twitter" href="https://twitter.com/mitocw"><img src="https://ocw.mit.edu/images/icon_twitter.png" alt="Click to visit our Twitter feed."></a><a aria-label="YouTube" href="https://www.youtube.com/mitocw" style="font-size: 12.208px;"><img src="https://ocw.mit.edu/images/icon_yt.png" alt="Click to visit our YouTube page."></a><span style="font-size: 12.208px;">   </span><a aria-label="Open Matters blog on WordPress" href="https://www.ocw-openmatters.org/" style="font-size: 12.208px;"><img src="https://ocw.mit.edu/images/icon_wp.png" alt="Click to visit our blog on WordPress."></a>
</td>
        </tr>
    </tbody>
</table>
</aside><nav aria-label="Help Links" class="helplinks">     <a aria-label="OCW Site Help" href="https://ocw.mit.edu/help">Help</a><span aria-hidden="true">|</span>     <a href="../../../common/about/contactus.htm">Contact Us</a>   </nav>
</div>
<div class="clear"> </div>
<!--googleon: index-->
</div>

</div>

<div id="portletwrapper-6f63772e746f70706f72746c65746d616e616765720a636f6e746578740a2f506c6f6e650a6d6567612d6d656e75" class="portletWrapper kssattr-portlethash-6f63772e746f70706f72746c65746d616e616765720a636f6e746578740a2f506c6f6e650a6d6567612d6d656e75">
<div class="portletStaticText portlet-static-mega-menu"><div><nav id="mega" class="grid_8 alpha" aria-label="Site">
<ul id="menu" role="presentation">
    <li id="menu_home">
<a href="https://ocw.mit.edu/" aria-label="Homepage"><img src="../../../common/images/top-nav_home.png" class="home_icon" alt="Click for site home page."></a><!-- Begin Home Item -->
</li>
    <!-- End Home Item -->
    <li id="drop_1" aria-label="Find Courses">
<a href="#" aria-hidden="true">FIND COURSES</a><!-- Begin 5 columns Item -->
    <div class="dropdown_5columns-a mega-courses">
    <div class="col_1a">
    <div class="row_1a">
<nav aria-labelledby="mm-find-courses-by">     <span id="mm-find-courses-by" class="nav" aria-hidden="true">Find courses by:</span>
    <ul class="find_by" role="presentation">
        <li><a href="https://ocw.mit.edu/courses/find-by-topic/">Topic</a></li>
        <li><a href="https://ocw.mit.edu/courses/find-by-number/">MIT Course Number</a></li>
        <li><a href="https://ocw.mit.edu/courses/find-by-department/">Department</a></li>
    </ul>
    </nav>     <nav aria-labelledby="mm-collections">     <span id="mm-collections" class="nav" aria-hidden="true">Collections</span>
    <ul role="presentation">
        <li><a href="https://ocw.mit.edu/courses/new-courses/">New Courses</a></li>
        <li><a href="https://ocw.mit.edu/courses/most-visited-courses/">Most Visited Courses</a></li>
        <li><a href="https://ocw.mit.edu/courses/ocw-scholar/">OCW Scholar Courses</a></li>
        <li><a href="https://ocw.mit.edu/courses/audio-video-courses/">Audio/Video Lectures</a></li>
        <li><a href="https://ocw.mit.edu/courses/online-textbooks/">Online Textbooks</a></li>
        <li><a href="https://ocw.mit.edu/resources/">Supplemental Resources</a></li>
        <li><a href="https://ocw.mit.edu/high-school/">OCW Highlights for High School</a></li>
        <li><a href="https://ocw.mit.edu/courses/mitx-related-courseware/">MITx &amp; Related OCW Courses</a></li>
        <li><a href="https://ocw.mit.edu/courses/mit-open-learning-library/">MIT Open Learning Library</a></li>
    </ul>
    </nav>     <nav class="col_1b" aria-labelledby="mm-translated-courses">     <span id="mm-translated-courses" class="nav" aria-hidden="true" style="line-height: 1.3;">Cross-Disciplinary Topic Lists</span>
    <ul role="presentation">
        <li><a href="https://ocw.mit.edu/courses/energy-courses">Energy</a></li>
        <li><a href="https://ocw.mit.edu/courses/entrepreneurship">Entrepreneurship</a></li>
        <li><a href="https://ocw.mit.edu/courses/environment-courses">Environment</a></li>
        <li><a href="https://ocw.mit.edu/courses/intro-programming">Introductory Programming</a></li>
        <li><a href="https://ocw.mit.edu/courses/life-sciences">Life Sciences</a></li>
        <li><a href="https://ocw.mit.edu/courses/transportation-courses">Transportation</a></li>
    </ul>
    </nav>
</div>
    <div class="row_1b"><nav aria-labelledby="mm-cross-disciplinary-topic-lists">     <span id="mm-cross-disciplinary-topic-lists" class="nav" aria-hidden="true">Translated Courses</span>
    <ul role="presentation">
        <li><a href="https://ocw.mit.edu/courses/translated-courses/traditional-chinese" aria-label="Traditional Chinese">繁體字 / Traditional Chinese</a></li>
        <li><a href="https://ocw.mit.edu/courses/translated-courses/turkish" aria-label="Turkish">Türkçe / Turkish</a></li>
        <li><a href="https://ocw.mit.edu/courses/translated-courses/korean" aria-label="Korean">(비디오)한국 / Korean</a></li>
    </ul>
    </nav></div>
    </div>
    </div>
    </li>
    <li id="drop_2">
<a href="#" aria-label="For Educators">For Educators</a>
    <div class="dropdown_1column-a" style="width: 270px;"><nav aria-labelledby="mm-find-courses-by">
    <ul role="presentation">
        <li><a href="https://ocw.mit.edu/educator/chalk-radio-podcast">Chalk Radio Podcast</a></li>
        <li><a href="https://ocw.mit.edu/educator/">OCW Educator Portal</a></li>
    </ul>
    <ul role="presentation">
        <li><a href="https://ocw.mit.edu/courses/instructor-insights/">Instructor Insights by Department</a></li>
        <li><a href="https://openlearning.mit.edu/campus/digital-innovations/">Residential Digital Innovations </a></li>
    </ul>
    <ul role="presentation">
        <li><a href="https://ocw.mit.edu/high-school/">OCW Highlights for High School</a></li>
    </ul>
    <ul role="presentation">
        <li><a href="https://ocw.mit.edu/educator/additional-resources/">Additional Resources</a></li>
    </ul>
    </nav></div>
    </li>
    <li id="drop_3">
<a href="#" aria-hidden="true">Give Now</a>
    <div class="dropdown_1column-a"><nav class="col_1" aria-label="Donate">
    <ul role="presentation">
        <li><a href="https://ocw.mit.edu/give/">Make a Donation</a></li>
        <li><a href="https://ocw.mit.edu/give/why-give/">Why Give?</a></li>
        <li><a href="https://ocw.mit.edu/give/our-supporters/">Our Supporters</a></li>
        <li><a href="https://ocw.mit.edu/give/other-ways-to-contribute/">Other Ways to Contribute</a></li>
        <li><a href="https://ocw.mit.edu/support/">Become a Corporate Sponsor</a></li>
    </ul>
    </nav></div>
    </li>
    <li id="drop_4">
<a href="#" aria-hidden="true">About</a>
    <div class="dropdown_1column-a"><nav class="col_1" aria-label="About">
    <ul role="presentation">
        <li><a href="https://ocw.mit.edu/about/">About MIT OpenCourseWare</a></li>
        <li><a href="https://ocw.mit.edu/about/site-statistics/">Site Statistics</a></li>
        <li><a href="https://ocw.mit.edu/about/ocw-stories/">OCW Stories</a></li>
        <li><a href="https://ocw.mit.edu/about/newsletter/">Newsletter</a></li>
        <li><a href="https://chalk-radio.simplecast.com/">Chalk Radio Podcast</a></li>
        <li><a href="https://www.ocw-openmatters.org/">Open Matters Blog</a></li>
    </ul>
    </nav></div>
    </li>
</ul>
</nav></div></div>

</div>





<!--googleoff: index-->
<script>
  (function() {
	var cx = '012626166551961672889:owjdpuboktq';
	var gcse = document.createElement('script');
	gcse.type = 'text/javascript';
	gcse.async = true;
	gcse.src = 'https://cse.google.com/cse.js?cx=' + cx;
	var s = document.getElementsByTagName('script')[0];
	s.parentNode.insertBefore(gcse, s);
  })();
  window.onload = function(){
	document.getElementById('gsc-i-id1').placeholder = 'Search';	
  };

$(document).ready(function(){

$('.advanceSearch a').keydown(function(event){showSearchTips($(this),event);})
$('#searchTipsModal').keydown(function(event){showSearchTips($(this),event);})

function showSearchTips(obj,evt) {
// if pressed enter key
	if ( evt.which == 13 || evt.which == 32) {
		showModal();
		ga('send', 'pageview', "AdvanceSearch");
		$(".advanceSearch a").attr('aria-expanded', 'true');
		$('.searchTipsModal').focus();
		evt.preventDefault();
		}
	if ( evt.which == 27 ) {
		hideModal();
		$(".advanceSearch a").attr('aria-expanded', 'false');
		var modal = document.getElementById('searchTipsModal');
		modal.style.display = "none";
		$(".advanceSearch a").focus();
		evt.preventDefault();
		}
}

});
function showModal(){
	var modal = document.getElementById('searchTipsModal');
	modal.style.display = "block";
	ga('send', 'pageview', "AdvanceSearch");
	document.getElementById("searchTipsBtn").setAttribute("aria-expanded", true);
	window.setTimeout(function () {
    document.getElementById('searchTipsModal').focus(); }, 0);
}

function hideModal(){
	var modal = document.getElementById('searchTipsModal');
	modal.style.display = "none";
	document.getElementById("searchTipsBtn").setAttribute("aria-expanded", false);
}
</script>
<div id="search" role="search" class="grid_4 omega">
    	<table class="search">
				<tbody>
					<tr>
						<td><div class="searchboxheader"><searchbox-only resultsurl="/search/ocwsearch.htm"></searchbox-only></div></td>
						<td>
							<div class="advanceSearch">
								<a id="searchTipsBtn" onclick="showModal();" href="#" role="button" aria-label="search tips" aria-expanded="false" aria-describedby="searchtips">Search Tips</a>

								<!-- The Modal -->
								<div id="searchTipsModal" class="modal" tabindex="-1">
								  <!-- Modal content -->
									<div class="modal-content">
										<div class="modal-body">
											<button class="close" onclick="hideModal();" aria-label="close">X</button>
											<span>
												<b>Exclude words from your search</b>
												<br>Put - in front of a word you want to leave out. For example, jaguar speed -car
												<br><br>
												<b>Search for an exact match</b>
												<br>Put a word or phrase inside quotes. For example, "tallest building".
												<br><br>
												<b>Search for wildcards or unknown words</b>
												<br>Put a * in your word or phrase where you want to leave a placeholder. For example, "largest * in the world".
												<br><br>
												<b>Search within a range of numbers</b>
												<br>Put .. between two numbers. For example, camera $50..$100.
												<br><br>
												<b>Combine searches</b>
												<br>Put "OR" between each search query. For example,  marathon OR race.
												<br><br>
											</span>
										</div>
								  </div>
								</div>
							</div>
						</td>
					</tr>
				</tbody>
		</table>
</div>
<div class="clear"></div>
<!--googleon: index-->
<!-- *end header* -->

				
				
			</div>
<!-- top grid end -->
		</header><!-- top end -->
			
		<div id="center_media">
      	<div id="grid">
      		<div id="left">
        		<nav id="breadcrumb_media" aria-label="Breadcrumb">
                	<p>

    <a href="https://ocw.mit.edu/">Home</a>
    
        »
        
    
    
        
            <a href="https://ocw.mit.edu/courses">Courses</a>
            
                »
                
            
            
         
    
    
        
            <a href="https://ocw.mit.edu/courses/electrical-engineering-and-computer-science">Electrical Engineering and Computer Science</a>
            
                »
                
            
            
         
    
    
        
            <a href="../../../contents/index.htm">Mathematics for Computer Science</a>
            
                »
                
            
            
         
    
    
        
            <a href="../../../contents/video-lectures/index.htm">Video Lectures</a>
            
                »
                
            
            
         
    
    
        
            
            
            Lecture 8: Graph Theory II: Minimum Spanning Trees
         
    
</p>

            	</nav>
            	<div class="clear"></div>
        		<div id="media_title">
        		<h1 class="title" itemprop="name" property="dct:title">
        <span class="" id="parent-fieldname-title">
            Lecture 8: Graph Theory II: Minimum Spanning Trees
        </span>
    </h1>
        		</div>
           		<div class="clear"></div>
           		<div id="course_wrapper_media">
           			<nav id="course_nav" aria-label="Course">
           				<script language="javascript" type="text/javascript">
function toggleMenu(objID) {
  if (!document.getElementById) return;
  var ob = document.getElementById(objID);
  ob.className = (ob.className == 'selected')?'': 'selected';
}
function toggleClass(id)
{
  var divtoggleClass= document.getElementById(id);
  divtoggleClass.className = (divtoggleClass.className == 'mO')?'mC': 'mO';
  return false;
}
function changeAlt(id)
{
  id.alt = (id.alt == 'Expand Menu')?'Collapse Menu' : 'Expand Menu';
  id.title = (id.title == 'Expand Menu')?'Collapse Menu' : 'Expand Menu';
}
</script>
<!--Left Nav Starts -->


	<ul>			  
	
	    	
	    	    <li class="">
			   			<a href="../../../contents/index.htm">
		                  Course Home  			                
	                    </a>
		        </li>
		    
         	
	
	
	    	
	    	    <li class="">
			   			<a href="../../../contents/syllabus/index.htm">
		                  Syllabus  			                
	                    </a>
		        </li>
		    
         	
	
	
	    	
	    	    <li class="">
			   			<a href="../../../contents/calendar/index.htm">
		                  Calendar  			                
	                    </a>
		        </li>
		    
         	
	
	
	    	
	    	    <li class="">
			   			<a href="../../../contents/readings/index.htm">
		                  Readings  			                
	                    </a>
		        </li>
		    
         	
	
	
	    	
	    	    <li class="selected">
			   			<a href="../../../contents/video-lectures/index.htm">
		                  Video Lectures  			                
	                    </a>
		        </li>
		    
         	
	
	
	    	
	    	    <li class="">
			   			<a href="../../../contents/recitations/index.htm">
		                  Recitations  			                
	                    </a>
		        </li>
		    
         	
	
	
	    	
	    	    <li class="">
			   			<a href="../../../contents/assignments/index.htm">
		                  Assignments  			                
	                    </a>
		        </li>
		    
         	
	
	
	    	
	    	    <li class="">
			   			<a href="../../../contents/exams/index.htm">
		                  Exams  			                
	                    </a>
		        </li>
		    
         	
	
	
	    	
	    	    
		    
         	
	<!--second tal block close-->  
	
</ul>


<!--Left Nav Ends -->





           			</nav>
           			<main id="course_inner_media" aria-labelledby="media_title">
      					 
        <div class="" id="parent-fieldname-text">
            
            
        </div>
    
      					     
    
    



<script type="text/javascript">var caption_embed_1 ={'English - US': '/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-8-graph-theory-ii-minimum-spanning-trees/GJpt_3ie4WU.srt'}</script>     
     <div id="media-embed">
         <div class="attention_message" id="embed_1">
<p>Flash and JavaScript are required for this feature.</p>
<p>Download the video from <a href="http://itunes.apple.com/us/itunes-u/lecture-8-graph-theory-ii/id503873536?i=110644969">iTunes U</a> or the <a href="http://www.archive.org/download/MIT6.042JF10/MIT6_042JF10_lec08_300k.mp4">Internet Archive</a>.</p>
</div>
     </div>
    
     <script type="text/javascript">ocw_embed_chapter_media('embed_1', 'https://www.youtube.com/v/GJpt_3ie4WU', 'youtube', '/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-8-graph-theory-ii-minimum-spanning-trees', 'https://img.youtube.com/vi/GJpt_3ie4WU/0.jpg',0,0, 'https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-8-graph-theory-ii-minimum-spanning-trees/GJpt_3ie4WU.srt')</script>
	 
	 
	 		
	 		<div id="transcript1"></div>
				 <script type="text/javascript">createThreePlayParams(2, 781483, "embed_1", 0, 0)</script>
			 
	

     <div id="media_resource_next_prev_nav" style="margin-top: 1em;">
        <p>
        
            <a href="../../../contents/video-lectures/lecture-7-matching-problems/index.htm">
                <img src="../../../common/images/btn_previous_resource.png" style="margin: 0 30px 0 50px;" alt="Previous track" title="Previous track"></a>
     	
     	
        
            <a href="../../../contents/video-lectures/lecture-9-communication-networks/index.htm"> 
                <img src="../../../common/images/btn_next_resource.png" alt="Next track" title="Next track"></a>
       
       </p>
     </div>
 


<script type="text/javascript">
		window.onload=function(){
		init();
		
		}
		var tabLinks = new Array();
		var contentDivs = new Array();
		function init() {
		  // Grab the tab links and content divs from the page
		  var tabListItems = document.getElementById('tabs').childNodes;
		  for ( var i = 0; i < tabListItems.length; i++ ) {
			if ( tabListItems[i].nodeName == "LI" ) {
			  var tabLink = getFirstChildWithTagName( tabListItems[i], 'A' );
			  var id = getHash( tabLink.getAttribute('href') );
			  tabLinks[id] = tabLink;
			  contentDivs[id] = document.getElementById( id );
			}
		  }
		  // Assign onclick events to the tab links, and
		  // highlight the first tab
		  var i = 0;
		  for ( var id in tabLinks ) {
			tabLinks[id].onclick = showTab;
			tabLinks[id].onfocus = function() { this.blur() };
			if ( i == 0 ) tabLinks[id].className = 'selected';
			i++;
		  }
		  // Hide all content divs except the first
		  var i = 0;
		  for ( var id in contentDivs ) {
			if ( i != 0 ) contentDivs[id].className = 'tabContent hide';
			i++;
		  }
		}
		function showTab() {
		  var selectedId = getHash( this.getAttribute('href') );
		  // Highlight the selected tab, and dim all others.
		  // Also show the selected content div, and hide all others.
		  for ( var id in contentDivs ) {
			if ( id == selectedId ) {
			  tabLinks[id].className = 'selected';
			  contentDivs[id].className = 'tabContent';
			} else {
			  tabLinks[id].className = '';
			  contentDivs[id].className = 'tabContent hide';
			}
		  }
		  // Stop the browser following the link
		  return false;
		}
		function getFirstChildWithTagName( element, tagName ) {
		  for ( var i = 0; i < element.childNodes.length; i++ ) {
			if ( element.childNodes[i].nodeName == tagName ) return element.childNodes[i];
		  }
		}
		function getHash( url ) {
		  var hashPos = url.lastIndexOf ( '#' );
		  return url.substring( hashPos + 1 );
		}
 </script>	
 

  <div id="media_tabs">
     
        <ul id="tabs">
            <li class="first">
                <a href="#vid_about" class="selected">About this Video</a>
            </li>
            <li class="">
                <a href="#vid_index" class="">Playlist</a>
            </li>
            <li class="">
                <a href="#vid_playlist" class="">Transcript</a>
            </li>
            <li class="">
                <a href="#vid_related" class="">Download this Video</a>
            </li>
        </ul>
   
        <div id="vid_about" itemprop="description" class="tabContent">
<p><strong>Description:</strong> Explores the various measures of connectivity of graphs and how these can be used to categorize and analyze graphs.</p> <p><strong>Speaker:</strong> Marten van Dijk</p>
</div>
        <div id="vid_index" itemprop="description" class="tabContent hide">
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-1-introduction-and-proofs/index.htm">
<img src="https://img.youtube.com/vi/L3LMbpZIKhQ/default.jpg" title="Lecture 1: Introduction and Proofs" alt="Lecture 1: Introduction and Proofs">
<p>Lecture 1: Introduction and...</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-2-induction/index.htm">
<img src="https://img.youtube.com/vi/z8HKWUWS-lA/default.jpg" title="Lecture 2: Induction" alt="Lecture 2: Induction">
<p>Lecture 2: Induction</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-3-strong-induction/index.htm">
<img src="https://img.youtube.com/vi/NuGDkmwEObM/default.jpg" title="Lecture 3: Strong Induction" alt="Lecture 3: Strong Induction">
<p>Lecture 3: Strong Induction</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-4-number-theory-i/index.htm">
<img src="https://img.youtube.com/vi/NuY7szYSXSw/default.jpg" title="Lecture 4: Number Theory I" alt="Lecture 4: Number Theory I">
<p>Lecture 4: Number Theory I</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-5-number-theory-ii/index.htm">
<img src="https://img.youtube.com/vi/XX7ePR21Ook/default.jpg" title="Lecture 5: Number Theory II" alt="Lecture 5: Number Theory II">
<p>Lecture 5: Number Theory II</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-6-graph-theory-and-coloring/index.htm">
<img src="https://img.youtube.com/vi/h9wxtqoa1jY/default.jpg" title="Lecture 6: Graph Theory and Coloring" alt="Lecture 6: Graph Theory and Coloring">
<p>Lecture 6: Graph Theory and...</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-7-matching-problems/index.htm">
<img src="https://img.youtube.com/vi/5RSMLgy06Ew/default.jpg" title="Lecture 7: Matching Problems" alt="Lecture 7: Matching Problems">
<p>Lecture 7: Matching Problems</p></a>
</div>
<div class="related-media-thumbnail-nolink">
<div class="now-playing-resource">Now Playing</div>
<img src="https://img.youtube.com/vi/GJpt_3ie4WU/default.jpg" title="Lecture 8: Graph Theory II: Minimum Spanning Trees" alt="Lecture 8: Graph Theory II: Minimum Spanning Trees">
<p>Lecture 8: Graph Theory II:...</p>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-9-communication-networks/index.htm">
<img src="https://img.youtube.com/vi/bTyxpoi2dmM/default.jpg" title="Lecture 9: Communication Networks" alt="Lecture 9: Communication Networks">
<p>Lecture 9: Communication Ne...</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-10-graph-theory-iii/index.htm">
<img src="https://img.youtube.com/vi/DOIp5D7VMS4/default.jpg" title="Lecture 10: Graph Theory III" alt="Lecture 10: Graph Theory III">
<p>Lecture 10: Graph Theory III</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-11-relations-partial-orders-and-scheduling/index.htm">
<img src="https://img.youtube.com/vi/1nScXLQAQ9A/default.jpg" title="Lecture 11: Relations, Partial Orders, and Scheduling" alt="Lecture 11: Relations, Partial Orders, and Scheduling">
<p>Lecture 11: Relations, Part...</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-12-sums/index.htm">
<img src="https://img.youtube.com/vi/fAeShezAGLE/default.jpg" title="Lecture 12: Sums" alt="Lecture 12: Sums">
<p>Lecture 12: Sums</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-13-sums-and-asymptotics/index.htm">
<img src="https://img.youtube.com/vi/X9eErxRjQEI/default.jpg" title="Lecture 13: Sums and Asymptotics" alt="Lecture 13: Sums and Asymptotics">
<p>Lecture 13: Sums and Asympt...</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-14-divide-and-conquer-recurrences/index.htm">
<img src="https://img.youtube.com/vi/Kqf0uO0oV6s/default.jpg" title="Lecture 14: Divide and Conquer Recurrences" alt="Lecture 14: Divide and Conquer Recurrences">
<p>Lecture 14: Divide and Conq...</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-15-linear-recurrences/index.htm">
<img src="https://img.youtube.com/vi/TWBB-JlmYUc/default.jpg" title="Lecture 15: Linear Recurrences" alt="Lecture 15: Linear Recurrences">
<p>Lecture 15: Linear Recurrences</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-16-counting-rules-i/index.htm">
<img src="https://img.youtube.com/vi/pNt5Ll6hGqo/default.jpg" title="Lecture 16: Counting Rules I" alt="Lecture 16: Counting Rules I">
<p>Lecture 16: Counting Rules I</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-17-counting-rules-ii/index.htm">
<img src="https://img.youtube.com/vi/09yIb3VHhMI/default.jpg" title="Lecture 17: Counting Rules II" alt="Lecture 17: Counting Rules II">
<p>Lecture 17: Counting Rules II</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-18-probability-introduction/index.htm">
<img src="https://img.youtube.com/vi/SmFwFdESMHI/default.jpg" title="Lecture 18: Probability Introduction" alt="Lecture 18: Probability Introduction">
<p>Lecture 18: Probability Int...</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-19-conditional-probability/index.htm">
<img src="https://img.youtube.com/vi/E6FbvM-FGZ8/default.jpg" title="Lecture 19: Conditional Probability" alt="Lecture 19: Conditional Probability">
<p>Lecture 19: Conditional Pro...</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-20-independence/index.htm">
<img src="https://img.youtube.com/vi/l1BCv3qqW4A/default.jpg" title="Lecture 20: Independence" alt="Lecture 20: Independence">
<p>Lecture 20: Independence</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-21-random-variables/index.htm">
<img src="https://img.youtube.com/vi/MOfhhFaQdjw/default.jpg" title="Lecture 21: Random Variables" alt="Lecture 21: Random Variables">
<p>Lecture 21: Random Variables</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-22-expectation-i/index.htm">
<img src="https://img.youtube.com/vi/gGlMSe7uEkA/default.jpg" title="Lecture 22: Expectation I" alt="Lecture 22: Expectation I">
<p>Lecture 22: Expectation I</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-23-expectation-ii/index.htm">
<img src="https://img.youtube.com/vi/oI9fMUqgfxY/default.jpg" title="Lecture 23: Expectation II" alt="Lecture 23: Expectation II">
<p>Lecture 23: Expectation II</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-24-large-deviations/index.htm">
<img src="https://img.youtube.com/vi/q4mwO2qS2z4/default.jpg" title="Lecture 24: Large Deviations" alt="Lecture 24: Large Deviations">
<p>Lecture 24: Large Deviations</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-25-random-walks/index.htm">
<img src="https://img.youtube.com/vi/56iFMY8QW2k/default.jpg" title="Lecture 25: Random Walks" alt="Lecture 25: Random Walks">
<p>Lecture 25: Random Walks</p></a>
</div>
</div>
        <div id="vid_playlist" itemprop="description" class="tabContent hide">
<ul><li><a class="transcript-link" title="Open in a new window." alt="Open in a new window." style="text-decoration: none; font-size: 1.0em;" target="_blank" text-decoration: none font-size: href="../../../contents/video-lectures/lecture-8-graph-theory-ii-minimum-spanning-trees/GJpt_3ie4WU.pdf"> Download English-US transcript (PDF)</a></li></ul>
<p><span m="410">The</span> <span m="510">following</span> <span m="950">content</span> <span m="1550">is</span> <span m="1660">provided</span> <span m="2110">under</span> <span m="2390">a</span> <span m="2420">Creative</span> <span m="2830">Commons</span> <span m="3230">license.</span> <span m="4340">Your</span> <span m="4530">support</span> <span m="5040">will</span> <span m="5200">help</span> <span m="5440">MIT</span> <span m="5890">OpenCourseWare</span> <span m="6680">continue</span> <span m="7190">to</span> <span m="7270">offer</span> <span m="7690">high</span> <span m="7920">quality</span> <span m="8450">educational</span> <span m="9070">resources</span> <span m="9700">for</span> <span m="9850">free.</span> <span m="11050">To</span> <span m="11060">make</span> <span m="11260">a</span> <span m="11310">donation</span> <span m="11990">or</span> <span m="12260">view</span> <span m="12700">additional</span> <span m="13130">materials</span> <span m="13670">from</span> <span m="13810">hundreds</span> <span m="14250">of</span> <span m="14360">MIT</span> <span m="14780">courses,</span> <span m="15900">visit</span> <span m="16110">MIT</span> <span m="16530">OpenCourseWare</span> <span m="17580">at</span> <span m="17740">ocw.mit.edu.</span> </p>
<p></p>
<p><span m="23780">PROFESSOR: So</span> <span m="24080">welcome</span> <span m="24400">to</span> <span m="24510">this</span> <span m="24730">week.</span> <span m="26300">We're</span> <span m="26490">going</span> <span m="26700">to</span> <span m="26830">talk</span> <span m="27120">about</span> <span m="28390">trees</span> <span m="28760">mainly.</span> <span m="29190">We're</span> <span m="29590">going</span> <span m="29780">to</span> <span m="29880">talk</span> <span m="30120">about</span> <span m="30320">some</span> <span m="30420">very</span> <span m="30640">special</span> <span m="31090">ones,</span> <span m="31640">spanning</span> <span m="32110">trees.</span> <span m="33770">But</span> <span m="34190">before</span> <span m="34480">we</span> <span m="34580">do</span> <span m="34730">this,</span> <span m="35370">you</span> <span m="35600">will go</span> <span m="35770">through</span> <span m="35970">a</span> <span m="36030">whole</span> <span m="36250">bunch</span> <span m="36510">of</span> <span m="36610">definitions</span> <span m="37690">and</span> <span m="37880">look</span> <span m="38040">at</span> <span m="38160">examples</span> <span m="39780">to</span> <span m="39920">explain</span> <span m="40340">how</span> <span m="40510">it</span> <span m="40650">works.</span> </p>
<p><span m="45910">So</span> <span m="45980">let's</span> <span m="46210">talk</span> <span m="46440">about</span> <span m="47220">walks</span> <span m="47660">and</span> <span m="48090">paths.</span> </p>
<p></p>
<p><span m="54290">So</span> <span m="54480">you've</span> <span m="54710">already</span> <span m="55010">seen</span> <span m="56662">a</span> <span m="57160">bit</span> <span m="57340">of</span> <span m="57470">graph</span> <span m="58080">theory.</span> <span m="58450">We've</span> <span m="58680">talked</span> <span m="59000">about</span> <span m="60410">graph</span> <span m="60770">colorings</span> <span m="61510">and</span> <span m="61700">so</span> <span m="61900">on.</span> </p>
<p><span m="63000">But</span> <span m="63120">now we're</span> <span m="63410">going</span> <span m="63610">to</span> <span m="63710">talk</span> <span m="64019">about</span> <span m="64569">special</span> <span m="65150">types</span> <span m="65480">of</span> <span m="65610">graphs,</span> <span m="66560">and</span> <span m="66730">special</span> <span m="67430">structures</span> <span m="67725">within</span> <span m="68020">such</span> <span m="68510">graphs.</span> </p>
<p></p>
<p><span m="72438">Well,</span> <span m="72864">let's</span> <span m="73290">start</span> <span m="73730">with the</span> <span m="73790">first</span> <span m="74060">definition.</span> <span m="75510">How</span> <span m="75670">do</span> <span m="75760">we</span> <span m="75880">define</span> <span m="76210">a</span> <span m="76280">walk?</span> <span m="76850">Well, a</span> <span m="77050">walk is</span> <span m="77340">something</span> <span m="77690">very</span> <span m="77900">simple.</span> <span m="78380">You</span> <span m="78850">walk</span> <span m="79280">from</span> <span m="79940">one</span> <span m="80760">vertex</span> <span m="81800">in</span> <span m="81940">the</span> <span m="82030">graph</span> <span m="82610">over</span> <span m="82890">an</span> <span m="83010">edge</span> <span m="83250">to</span> <span m="83400">a</span> <span m="83510">next</span> <span m="83950">vertex,</span> <span m="84390">to</span> <span m="84520">a</span> <span m="84630">next</span> <span m="85080">vertex,</span> <span m="85440">and</span> <span m="85610">so</span> <span m="85820">on.</span> </p>
<p><span m="86740">So</span> <span m="86840">how</span> <span m="86970">do</span> <span m="87040">we</span> <span m="87150">define</span> <span m="87580">this?</span> <span m="89370">A</span> <span m="89490">walk</span> <span m="90830">is</span> <span m="92000">a</span> <span m="92100">sequence</span> <span m="97180">of</span> <span m="98020">vertices</span> <span m="98620">that</span> <span m="98900">are</span> <span m="99170">connected</span> <span m="105790">by</span> <span m="106290">edges.</span> <span m="107920">So</span> <span m="109050">for</span> <span m="109280">example,</span> <span m="109880">we</span> <span m="109910">may</span> <span m="110130">have</span> <span m="110300">something</span> <span m="110910">that</span> <span m="111080">looks</span> <span m="111400">like</span> <span m="111830">a</span> <span m="111950">first</span> <span m="112310">vertex.</span> <span m="113470">That</span> <span m="113960">we call the</span> <span m="114020">start.</span> <span m="115520">And</span> <span m="115940">an</span> <span m="116140">edge</span> <span m="116700">that</span> <span m="116870">goes</span> <span m="117130">to</span> <span m="117960">a</span> <span m="118050">second</span> <span m="118380">vertex.</span> <span m="120760">We</span> <span m="120910">continue</span> <span m="121360">like</span> <span m="121550">this</span> <span m="121950">until</span> <span m="122390">the end,</span> <span m="122860">say,</span> <span m="123370">at</span> <span m="123460">vk,</span> <span m="124800">which</span> <span m="124970">we</span> <span m="125110">call</span> <span m="125520">the</span> <span m="125670">end</span> <span m="126260">of</span> <span m="126400">the</span> <span m="126510">walk.</span> </p>
<p><span m="127520">And</span> <span m="129120">if</span> <span m="129330">you</span> <span m="129440">look</span> <span m="129720">at</span> <span m="129870">this,</span> <span m="130820">we</span> <span m="130949">say</span> <span m="131200">this</span> <span m="131370">is</span> <span m="131450">a</span> <span m="131580">walk</span> <span m="131800">from</span> <span m="131960">v0</span> <span m="132470">all</span> <span m="132640">the</span> <span m="132710">way</span> <span m="132850">to</span> <span m="133000">vk.</span> <span m="133900">It</span> <span m="134060">has</span> <span m="134260">exactly</span> <span m="134810">k</span> <span m="135160">edges.</span> <span m="136250">And</span> <span m="136380">we</span> <span m="136470">say</span> <span m="136700">that</span> <span m="136780">the</span> <span m="136890">length</span> <span m="139710">is</span> <span m="139940">actually</span> <span m="140330">equal</span> <span m="140520">to</span> <span m="140680">k.</span> </p>
<p><span m="143130">So</span> <span m="143300">this</span> <span m="143520">is</span> <span m="144420">a</span> <span m="144890">very</span> <span m="145180">basic</span> <span m="145640">notion.</span> <span m="146570">And</span> <span m="146740">what</span> <span m="146860">we</span> <span m="147240">are</span> <span m="147510">really</span> <span m="147730">interested</span> <span m="148220">in</span> <span m="148660">is</span> <span m="148850">actually</span> <span m="149900">paths.</span> <span m="151770">And</span> <span m="151960">paths</span> <span m="152400">are</span> <span m="153150">special</span> <span m="153660">types</span> <span m="153990">of</span> <span m="154070">walks</span> <span m="155620">in</span> <span m="155810">which</span> <span m="156190">all</span> <span m="156410">those</span> <span m="156720">vertices</span> <span m="158190">are</span> <span m="158370">actually</span> <span m="158700">different</span> <span m="159050">from</span> <span m="159200">one</span> <span m="159360">another.</span> <span m="160700">But</span> <span m="160850">let's</span> <span m="161430">first</span> <span m="162350">give</span> <span m="162550">an</span> <span m="162620">example</span> <span m="163200">of</span> <span m="164140">a</span> <span m="164230">graph</span> <span m="164550">that you will</span> <span m="164870">study</span> <span m="165210">throughout</span> <span m="165550">the</span> <span m="165610">whole</span> <span m="165790">lecture,</span> <span m="166550">especially</span> <span m="167030">when</span> <span m="167170">we</span> <span m="167280">come</span> <span m="167430">to</span> <span m="167530">spanning</span> <span m="167910">trees.</span> </p>
<p></p>
<p><span m="170860">So</span> <span m="171060">let's</span> <span m="171320">have</span> <span m="171550">a</span> <span m="171610">look</span> <span m="171970">at</span> <span m="172110">the</span> <span m="172190">following</span> <span m="174800">graph.</span> </p>
<p></p>
<p><span m="194290">So</span> <span m="194420">let</span> <span m="194740">this be a</span> <span m="194930">graph.</span> <span m="195840">And</span> <span m="195990">for</span> <span m="196120">example,</span> <span m="196890">we</span> <span m="196940">can</span> <span m="197110">have</span> <span m="197300">a</span> <span m="197420">walk</span> <span m="198110">that</span> <span m="198400">goes</span> <span m="198870">from</span> <span m="202770">this</span> <span m="202950">particular</span> <span m="203410">edge.</span> <span m="203890">Say</span> <span m="204120">we</span> <span m="204260">call</span> <span m="204490">this</span> <span m="205000">v0.</span> <span m="207060">It</span> <span m="207260">goes</span> <span m="207620">over</span> <span m="207870">this</span> <span m="208170">edge</span> <span m="208930">over</span> <span m="209260">here.</span> <span m="209640">Suppose</span> <span m="209980">I</span> <span m="210080">want</span> <span m="210200">to</span> <span m="210300">end</span> <span m="210570">over</span> <span m="210840">here.</span> </p>
<p><span m="212610">Well,</span> <span m="212770">I</span> <span m="212820">can</span> <span m="213300">go</span> <span m="213480">many</span> <span m="213760">ways</span> <span m="214050">in</span> <span m="214130">this</span> <span m="214340">particular</span> <span m="214790">one.</span> <span m="215030">I</span> <span m="215080">can</span> <span m="215360">go</span> <span m="215700">for</span> <span m="215870">this</span> <span m="216140">edge.</span> <span m="216435">I</span> <span m="216730">may</span> <span m="216980">go over</span> <span m="217450">here.</span> <span m="218670">I</span> <span m="218760">may</span> <span m="218940">go</span> <span m="219180">all</span> <span m="219400">the</span> <span m="219500">way</span> <span m="219680">to</span> <span m="219880">this</span> <span m="220190">part</span> <span m="220530">over</span> <span m="220800">here.</span> <span m="221040">I may</span> <span m="221200">return</span> <span m="221700">if</span> <span m="221850">I</span> <span m="221930">want</span> <span m="222220">to.</span> <span m="222760">And</span> <span m="222890">I</span> <span m="222920">finally,</span> <span m="223320">I</span> <span m="223360">take</span> <span m="223630">this</span> <span m="223910">edge</span> <span m="224100">for</span> <span m="224300">example</span> <span m="224780">back</span> <span m="225750">to--</span> <span m="227010">over</span> <span m="227270">this</span> <span m="227500">edge,</span> <span m="227640">I</span> <span m="227790">will</span> <span m="228170">end</span> <span m="228630">in</span> <span m="228950">the</span> <span m="229060">last</span> <span m="229580">vertex</span> <span m="229925">vk.</span> </p>
<p><span m="231490">So</span> <span m="232250">what</span> <span m="232410">we</span> <span m="232530">notice</span> <span m="232920">here</span> <span m="233140">is</span> <span m="233360">that</span> <span m="233520">we have a</span> <span m="233660">walk</span> <span m="234050">that</span> <span m="234550">actually</span> <span m="236770">gets</span> <span m="237010">to</span> <span m="237120">this</span> <span m="237370">particular</span> <span m="237750">vertex,</span> <span m="238690">and</span> <span m="238940">then</span> <span m="239980">returns</span> <span m="240370">to</span> <span m="240520">this</span> <span m="240840">vertex.</span> <span m="241370">So</span> <span m="241440">for</span> <span m="241550">a</span> <span m="241580">few</span> <span m="241840">other</span> <span m="242050">edges,</span> <span m="242690">and</span> <span m="242930">then</span> <span m="243200">goes all the</span> <span m="243470">way</span> <span m="243840">to</span> <span m="243990">vk.</span> </p>
<p><span m="245470">So</span> <span m="245740">if</span> <span m="245880">you</span> <span m="245960">talk</span> <span m="246180">about</span> <span m="246430">the</span> <span m="246540">path,</span> <span m="247510">then</span> <span m="247850">we</span> <span m="248060">really</span> <span m="248370">want</span> <span m="249660">all</span> <span m="249880">the</span> <span m="249960">difference</span> <span m="250300">vertices</span> <span m="250720">not to</span> <span m="250990">occur</span> <span m="251340">twice.</span> <span m="252920">So</span> <span m="253310">let's</span> <span m="253530">define</span> <span m="253930">this</span> <span m="254890">so</span> <span m="255190">the</span> <span m="255250">destination</span> <span m="255950">is</span> <span m="257610">that</span> <span m="257860">a</span> <span m="258310">path</span> <span m="259200">is</span> <span m="259450">actually</span> <span m="259950">a</span> <span m="260050">walk.</span> </p>
<p></p>
<p><span m="264580">Is a</span> <span m="264750">walk</span> <span m="267280">where</span> <span m="267940">all</span> <span m="268910">edges,</span> <span m="269810">where</span> <span m="270030">all</span> <span m="270260">vertices,</span> <span m="273380">vi's,</span> <span m="274840">are</span> <span m="275140">different.</span> </p>
<p></p>
<p><span m="280790">In</span> <span m="280930">this</span> <span m="281090">particular</span> <span m="281560">case--</span> <span m="282810">well</span> <span m="282990">for</span> <span m="283110">example,</span> <span m="283490">if</span> <span m="283630">you</span> <span m="283720">strip</span> <span m="284070">off</span> <span m="284280">this</span> <span m="284510">part</span> <span m="284790">over</span> <span m="285060">here,</span> <span m="285810">you</span> <span m="285920">will</span> <span m="286130">get</span> <span m="286230">a</span> <span m="286300">path</span> <span m="286720">from</span> <span m="286930">v0</span> <span m="287630">to</span> <span m="287830">here,</span> <span m="288400">to</span> <span m="288540">here,</span> <span m="288820">to</span> <span m="289030">here.</span> <span m="289950">And</span> <span m="290270">all</span> <span m="290730">these</span> <span m="291440">vertices</span> <span m="291840">on</span> <span m="292990">the</span> <span m="293090">path</span> <span m="293760">on</span> <span m="293940">this</span> <span m="294140">walk</span> <span m="294450">are</span> <span m="294680">different.</span> <span m="295220">And</span> <span m="295360">therefore</span> <span m="295690">it is</span> <span m="296130">a</span> <span m="296180">path.</span> </p>
<p><span m="298130">This</span> <span m="298620">also</span> <span m="298830">shows</span> <span m="299190">us</span> <span m="299290">something.</span> <span m="300830">It</span> <span m="301000">is</span> <span m="301150">possible</span> <span m="301660">to</span> <span m="302030">construct</span> <span m="302560">from a</span> <span m="303020">walk</span> <span m="303550">a</span> <span m="303860">path.</span> <span m="305140">As</span> <span m="305260">you</span> <span m="305400">can</span> <span m="305550">see,</span> <span m="306240">we</span> <span m="306340">started</span> <span m="306780">off</span> <span m="306960">with</span> <span m="307070">this</span> <span m="307230">particular</span> <span m="307650">walk.</span> <span m="308560">And</span> <span m="309620">we</span> <span m="309770">just</span> <span m="310910">deleted,</span> <span m="311510">essentially,</span> <span m="312560">a</span> <span m="312670">part</span> <span m="314060">where</span> <span m="314290">we</span> <span m="314620">sort</span> <span m="314880">of</span> <span m="315040">came</span> <span m="315270">back</span> <span m="315530">to</span> <span m="315640">the</span> <span m="315740">same</span> <span m="315980">vertex</span> <span m="316470">again.</span> <span m="317630">And</span> <span m="317900">when</span> <span m="318060">we</span> <span m="318260">delete</span> <span m="318500">all</span> <span m="318650">those</span> <span m="318890">kinds</span> <span m="319210">of</span> <span m="319970">parts,</span> <span m="321230">you will</span> <span m="321500">end</span> <span m="321690">up</span> <span m="321870">with</span> <span m="322030">a</span> <span m="322100">path.</span> </p>
<p><span m="322530">So</span> <span m="322710">can</span> <span m="322840">we</span> <span m="322940">formalize</span> <span m="323470">this</span> <span m="323670">a</span> <span m="323710">little</span> <span m="323930">bit</span> <span m="324110">better?</span> <span m="325660">Then</span> <span m="325780">we</span> <span m="325910">get</span> <span m="326110">the</span> <span m="326510">next</span> <span m="326850">lemma.</span> </p>
<p></p>
<p><span m="331060">I</span> <span m="331610">call</span> <span m="331800">this</span> <span m="332340">lemma</span> <span m="332700">one.</span> <span m="334630">And</span> <span m="334890">let's</span> <span m="335110">see</span> <span m="335180">whether</span> <span m="335320">we</span> <span m="335440">can</span> <span m="335610">prove</span> <span m="335950">this.</span> <span m="336700">So</span> <span m="337190">we</span> <span m="337310">want</span> <span m="337440">to</span> <span m="337510">show</span> <span m="337840">that</span> <span m="338060">if</span> <span m="338270">there</span> <span m="338470">exists</span> <span m="339200">a</span> <span m="339330">walk</span> <span m="342020">from,</span> <span m="342920">say,</span> <span m="342970">a</span> <span m="343750">vertex</span> <span m="343920">u</span> <span m="345230">to--</span> <span m="346195">well,</span> <span m="346530">maybe</span> <span m="346830">I</span> <span m="346850">should</span> <span m="347100">not</span> <span m="347690">use</span> <span m="348120">arrows.</span> <span m="349900">That's</span> <span m="350110">confusing--</span> <span m="351920">from</span> <span m="352220">u</span> <span m="352490">to</span> <span m="352760">v.</span> <span m="354650">Then</span> <span m="357030">there</span> <span m="357210">also</span> <span m="357490">exists</span> <span m="358320">a</span> <span m="358510">path</span> <span m="360820">from</span> <span m="361640">u</span> <span m="363340">to</span> <span m="364160">v.</span> </p>
<p><span m="366140">So</span> <span m="366370">how</span> <span m="366530">can</span> <span m="366670">we</span> <span m="366740">prove</span> <span m="367040">this?</span> <span m="367410">Do</span> <span m="367740">you</span> <span m="367940">have</span> <span m="368000">any</span> <span m="368255">ideas?</span> </p>
<p><span m="369590">So</span> <span m="371480">what</span> <span m="371630">kind</span> <span m="371810">of</span> <span m="371990">proof</span> <span m="372350">techniques</span> <span m="372740">can</span> <span m="372910">we</span> <span m="373290">use</span> <span m="373680">here?</span> <span m="375190">And</span> <span m="376250">maybe</span> <span m="377255">one</span> <span m="377650">of</span> <span m="377800">the</span> <span m="378840">principles</span> <span m="379155">that</span> <span m="379470">we</span> <span m="379580">have</span> <span m="379920">seen?</span> </p>
<p><span m="381396">AUDIENCE: [INAUDIBLE]</span> <span m="384840">but I</span> <span m="385332">have</span> <span m="385824">an</span> <span m="386316">idea</span> <span m="386808">about how</span> <span m="387300">you could show this.</span> <span m="388284">It's</span> <span m="388790">basically</span> <span m="389650">if</span> <span m="389790">you</span> <span m="389920">visit</span> <span m="391092">one</span> <span m="391440">vertex--</span> <span m="392130">let's</span> <span m="392600">you</span> <span m="392730">go</span> <span m="395650">1, 2, 3, 4, 5, 3, 6.</span> <span m="398650">You can</span> <span m="399150">walk from</span> <span m="399650">1</span> <span m="400150">to 6.</span> <span m="402132">Then</span> <span m="402626">we just</span> <span m="403120">take</span> <span m="403614">out the</span> <span m="404110">4,</span> <span m="405230">you</span> <span m="405730">just</span> <span m="406130">go</span> <span m="406440">1, 2, 3, 6.</span> </p>
<p><span m="407210">PROFESSOR: Right.</span> <span m="407440">So</span> <span m="407600">what</span> <span m="407770">did</span> <span m="407930">we</span> <span m="408030">do</span> <span m="408210">there?</span> <span m="408520">We</span> <span m="408630">essentially</span> <span m="409130">had</span> <span m="409340">a</span> <span m="409410">walk.</span> <span m="409760">And</span> <span m="409910">then</span> <span m="410170">we</span> <span m="410330">cut</span> <span m="410530">out</span> <span m="410740">a</span> <span m="410860">smaller</span> <span m="411270">parts</span> <span m="411700">here</span> <span m="412250">that</span> <span m="412580">recurs</span> <span m="412935">in a</span> <span m="413290">sense.</span> <span m="414480">And</span> <span m="416000">we</span> <span m="416160">have</span> <span m="416310">shortened</span> <span m="416970">the</span> <span m="417080">path,</span> <span m="417930">we have</span> <span m="418100">shortened</span> <span m="418500">the</span> <span m="418610">walk</span> <span m="419060">to</span> <span m="419290">a</span> <span m="419340">smaller</span> <span m="419740">path.</span> </p>
<p><span m="420950">So</span> <span m="422090">what</span> <span m="422230">we've</span> <span m="422310">used</span> <span m="422600">here</span> <span m="422780">is</span> <span m="422970">we</span> <span m="423070">can</span> <span m="423240">sort</span> <span m="423480">of</span> <span m="424960">take</span> <span m="425270">out</span> <span m="426720">parts</span> <span m="427200">of</span> <span m="427410">the</span> <span m="427510">walk</span> <span m="427740">just</span> <span m="427950">like</span> <span m="428100">we</span> <span m="428330">did</span> <span m="428500">over</span> <span m="428760">here</span> <span m="429850">until</span> <span m="431100">the</span> <span m="431200">walk gets</span> <span m="431550">shorter,</span> <span m="431830">and</span> <span m="431960">shorter,</span> <span m="432200">and</span> <span m="432330">shorter.</span> <span m="433440">So</span> <span m="433610">maybe</span> <span m="434310">one</span> <span m="434510">of</span> <span m="434760">the</span> <span m="434860">things</span> <span m="435210">that</span> <span m="435340">we</span> <span m="435450">may</span> <span m="435970">consider</span> <span m="436420">is</span> <span m="437470">a</span> <span m="437600">walk</span> <span m="438240">of</span> <span m="438660">shortest</span> <span m="439160">length.</span> </p>
<p></p>
<p><span m="442980">And</span> <span m="443440">we</span> <span m="443560">know</span> <span m="443720">that</span> <span m="443910">this</span> <span m="445020">is</span> <span m="445140">possible.</span> <span m="447860">Oops,</span> <span m="448100">sorry</span> <span m="448510">about that.</span> </p>
<p></p>
<p><span m="454770">So</span> <span m="455010">let's</span> <span m="455410">prove</span> <span m="455710">this.</span> <span m="456470">So</span> <span m="456770">for</span> <span m="457020">example,</span> <span m="460040">suppose</span> <span m="460440">there</span> <span m="460510">exists</span> <span m="461100">such</span> <span m="461440">a</span> <span m="461520">walk</span> <span m="462250">from</span> <span m="462700">u to</span> <span m="463160">v.</span> <span m="464120">Then</span> <span m="464500">we</span> <span m="464600">know</span> <span m="465040">by</span> <span m="465470">the</span> <span m="465770">ordering</span> <span m="466150">principle</span> <span m="467030">that</span> <span m="467530">there</span> <span m="467810">exists</span> <span m="467950">a walk</span> <span m="468310">of</span> <span m="468500">minimum</span> <span m="468790">length.</span> <span m="469900">So</span> <span m="470080">that's</span> <span m="470320">what we're</span> <span m="470620">going</span> <span m="470830">to</span> <span m="470960">use</span> <span m="471250">here.</span> </p>
<p><span m="471960">And</span> <span m="472200">then we're</span> <span m="472310">going</span> <span m="472580">to</span> <span m="473910">essentially</span> <span m="475220">go</span> <span m="476000">in</span> <span m="476150">the</span> <span m="476230">same</span> <span m="476520">direction</span> <span m="476940">as</span> <span m="477060">what</span> <span m="477220">you</span> <span m="477340">were</span> <span m="477420">talking</span> <span m="477790">about</span> <span m="478580">because</span> <span m="478850">we're</span> <span m="479000">going</span> <span m="479200">to</span> <span m="479300">show</span> <span m="480220">that</span> <span m="480540">it's</span> <span m="480750">not</span> <span m="481090">possible</span> <span m="481840">to</span> <span m="482010">delete</span> <span m="482510">anything</span> <span m="483680">more.</span> <span m="484890">Because</span> <span m="485130">otherwise,</span> <span m="485680">if</span> <span m="485800">you</span> <span m="485890">could</span> <span m="486040">do</span> <span m="486180">that,</span> <span m="486470">the</span> <span m="486570">walk</span> <span m="486780">would</span> <span m="486890">get</span> <span m="487070">shorter.</span> <span m="487530">And</span> <span m="487710">that's</span> <span m="487880">not</span> <span m="488030">possible,</span> <span m="488600">because</span> <span m="488950">by</span> <span m="489080">the well</span> <span m="489360">ordering</span> <span m="489680">principle,</span> <span m="493010">we</span> <span m="495680">simply</span> <span m="496130">consider</span> <span m="498360">that</span> <span m="500160">there</span> <span m="500510">exists</span> <span m="500930">a</span> <span m="501050">walk</span> <span m="501410">of</span> <span m="501740">minimal</span> <span m="502100">length</span> <span m="502580">that</span> <span m="502690">we</span> <span m="502950">are</span> <span m="503120">interested</span> <span m="503740">in.</span> </p>
<p></p>
<p><span m="510440">We</span> <span m="510560">will</span> <span m="510700">show</span> <span m="511070">that</span> <span m="511230">this</span> <span m="511430">particular</span> <span m="511880">walk</span> <span m="512299">is</span> <span m="512640">actually</span> <span m="512940">going to</span> <span m="513210">be</span> <span m="513360">a path.</span> <span m="514630">So</span> <span m="514850">let's</span> <span m="515049">see</span> <span m="515230">how</span> <span m="515380">we</span> <span m="515480">can</span> <span m="515580">do</span> <span m="515720">this.</span> </p>
<p><span m="516950">So</span> <span m="517030">let's</span> <span m="517620">do</span> <span m="518070">walk</span> <span m="518905">be</span> <span m="519320">equal</span> <span m="519809">to--</span> <span m="520559">well, it</span> <span m="520710">starts</span> <span m="521059">in</span> <span m="521150">v0.</span> <span m="521770">It's</span> <span m="521890">equal</span> <span m="522120">to</span> <span m="522250">u.</span> <span m="523450">There's</span> <span m="523650">an</span> <span m="523770">edge</span> <span m="524130">to,</span> <span m="524260">say,</span> <span m="524550">v1.</span> <span m="525200">It</span> <span m="525660">goes</span> <span m="525940">all</span> <span m="526200">the</span> <span m="526260">way</span> <span m="526460">up</span> <span m="526600">to</span> <span m="527310">vk,</span> <span m="529080">which</span> <span m="529400">is</span> <span m="530040">the</span> <span m="530160">last</span> <span m="530520">vertex,</span> <span m="531400">v.</span> </p>
<p><span m="533450">So</span> <span m="533630">what</span> <span m="533760">are we</span> <span m="533920">going</span> <span m="534090">to</span> <span m="534200">prove?</span> <span m="534410">We're</span> <span m="534500">going</span> <span m="534620">to</span> <span m="534760">prove</span> <span m="534990">that</span> <span m="535050">this</span> <span m="535220">is</span> <span m="535330">actually</span> <span m="535670">a path.</span> <span m="537140">And</span> <span m="538420">since</span> <span m="538680">this</span> <span m="538840">is</span> <span m="538970">a</span> <span m="539040">walk of</span> <span m="539360">minimal</span> <span m="539720">length,</span> <span m="540790">well</span> <span m="541060">suppose</span> <span m="541620">it is</span> <span m="541920">not</span> <span m="542700">a</span> <span m="542810">path.</span> <span m="544200">So</span> <span m="544460">we</span> <span m="544840">are</span> <span m="545010">going</span> <span m="545210">to</span> <span m="545310">use</span> <span m="545540">contradiction.</span> </p>
<p><span m="548795">Well</span> <span m="549270">if it's</span> <span m="549400">not</span> <span m="549570">a</span> <span m="549690">path,</span> <span m="550540">then</span> <span m="550720">by</span> <span m="550990">our</span> <span m="551190">definitions</span> <span m="551750">over</span> <span m="552130">here,</span> <span m="553030">there</span> <span m="553150">must</span> <span m="553500">be</span> <span m="553670">two</span> <span m="553900">vertices</span> <span m="554350">that are</span> <span m="554640">actually</span> <span m="554950">the</span> <span m="555040">same,</span> <span m="555550">just</span> <span m="555830">like</span> <span m="556070">in</span> <span m="556160">here.</span> <span m="557170">So</span> <span m="557520">we</span> <span m="557650">go</span> <span m="557870">from</span> <span m="558050">v0</span> <span m="558860">to,</span> <span m="559180">say,</span> <span m="559430">this</span> <span m="559720">particular</span> <span m="560390">vertex,</span> <span m="560590">or</span> <span m="560700">this</span> <span m="560930">one.</span> <span m="562270">And</span> <span m="562360">then</span> <span m="562460">we</span> <span m="562560">come</span> <span m="562730">back</span> <span m="562970">to</span> <span m="563080">this</span> <span m="563330">one,</span> <span m="563900">and</span> <span m="564060">then</span> <span m="564250">all</span> <span m="564410">the</span> <span m="564470">way</span> <span m="564600">to</span> <span m="564740">vk.</span> </p>
<p><span m="565220">So</span> <span m="565870">if</span> <span m="566050">it</span> <span m="566180">is</span> <span m="566320">not</span> <span m="566540">a</span> <span m="566650">path,</span> <span m="569020">then</span> <span m="569290">two</span> <span m="569530">of</span> <span m="569650">those</span> <span m="569950">must</span> <span m="570180">be</span> <span m="570350">equal</span> <span m="570530">to</span> <span m="570650">one</span> <span m="570830">another.</span> <span m="572150">And</span> <span m="572410">then</span> <span m="572630">we are</span> <span m="572770">going</span> <span m="573080">to</span> <span m="573630">use</span> <span m="573920">this</span> <span m="574120">trick.</span> <span m="574520">We're going</span> <span m="574890">to</span> <span m="574950">take</span> <span m="575240">out</span> <span m="575410">this</span> <span m="575620">part.</span> <span m="576000">We get</span> <span m="576210">the</span> <span m="576290">shorter</span> <span m="576740">walk.</span> <span m="578070">But</span> <span m="578250">that</span> <span m="578450">contradicts</span> <span m="580310">that</span> <span m="581000">our</span> <span m="581160">walk is</span> <span m="581570">minimal</span> <span m="581900">length.</span> <span m="582660">So</span> <span m="582820">that's</span> <span m="583010">sort</span> <span m="583170">of</span> <span m="583260">the</span> <span m="583350">proof</span> <span m="583620">technique</span> <span m="583990">that</span> <span m="584100">we're</span> <span m="584310">doing.</span> </p>
<p><span m="585680">OK.</span> <span m="585970">Let's</span> <span m="586440">do</span> <span m="586580">this.</span> </p>
<p></p>
<p><span m="592140">So</span> <span m="594150">first,</span> <span m="595140">let's</span> <span m="595420">consider</span> <span m="595930">a</span> <span m="596020">few</span> <span m="596380">cases.</span> <span m="597640">If</span> <span m="597860">k</span> <span m="598060">equals</span> <span m="598430">0,</span> <span m="599610">then</span> <span m="600280">that's</span> <span m="600590">pretty</span> <span m="600860">obvious.</span> <span m="601400">We</span> <span m="601470">have</span> <span m="601740">only</span> <span m="602390">a</span> <span m="602470">single</span> <span m="602850">vertex</span> <span m="603320">u</span> <span m="603730">that</span> <span m="603970">is connected</span> <span m="604570">by</span> <span m="604710">a</span> <span m="604880">0</span> <span m="605290">length</span> <span m="605640">path</span> <span m="606100">to</span> <span m="606250">itself.</span> <span m="607500">So</span> <span m="607560">this</span> <span m="607760">is</span> <span m="607920">fine.</span> </p>
<p><span m="609230">For</span> <span m="609740">k</span> <span m="609920">equals</span> <span m="610340">1,</span> <span m="610920">we</span> <span m="611040">have</span> <span m="611300">just</span> <span m="611710">a</span> <span m="611780">single</span> <span m="612190">edge.</span> <span m="613310">And</span> <span m="613450">then</span> <span m="613530">we're</span> <span m="613640">done</span> <span m="613880">too.</span> <span m="614020">That's</span> <span m="614270">a</span> <span m="614620">path as</span> <span m="614770">well.</span> </p>
<p><span m="615830">So</span> <span m="616050">let's</span> <span m="617370">consider</span> <span m="617820">the</span> <span m="618000">case</span> <span m="618710">where</span> <span m="619350">k</span> <span m="619690">is</span> <span m="620000">at</span> <span m="620120">least</span> <span m="620530">two.</span> <span m="622230">Well,</span> <span m="623050">suppose</span> <span m="623540">it's</span> <span m="623670">a walk.</span> <span m="624080">It's not</span> <span m="624300">a path.</span> <span m="624540">So</span> <span m="624800">we</span> <span m="625400">are</span> <span m="625540">going</span> <span m="625880">to</span> <span m="627870">assume</span> <span m="628200">the</span> <span m="628350">opposite.</span> </p>
<p></p>
<p><span m="634310">If</span> <span m="634490">it's</span> <span m="634640">not</span> <span m="634780">a</span> <span m="634870">path,</span> <span m="636920">then</span> <span m="637140">we</span> <span m="637220">concluded</span> <span m="638340">that</span> <span m="638670">two</span> <span m="638930">of</span> <span m="639070">those</span> <span m="639340">vi's</span> <span m="639900">must</span> <span m="640100">be</span> <span m="640230">equal</span> <span m="640430">to</span> <span m="640560">one</span> <span m="640720">another.</span> <span m="642330">So</span> <span m="642430">there</span> <span m="642670">exits</span> <span m="643090">an</span> <span m="643380">i</span> <span m="643670">unequal</span> <span m="644060">to</span> <span m="644230">j</span> <span m="645250">such</span> <span m="645565">that</span> <span m="646320">vertex</span> <span m="646720">vi</span> <span m="647290">equals</span> <span m="648720">vj.</span> </p>
<p><span m="651208">Ah,</span> <span m="651660">but</span> <span m="651810">now</span> <span m="651980">we</span> <span m="652080">have</span> <span m="652490">a</span> <span m="652880">walk</span> <span m="653540">that</span> <span m="655290">goes</span> <span m="655800">from</span> <span m="656040">v0</span> <span m="658030">all</span> <span m="658290">the</span> <span m="658370">way</span> <span m="658570">to,</span> <span m="658770">say,</span> <span m="659190">vi,</span> <span m="660180">which</span> <span m="660320">is</span> <span m="660440">equal</span> <span m="660720">to</span> <span m="660890">vj.</span> <span m="662280">And</span> <span m="662510">then</span> <span m="662660">we</span> <span m="662740">go</span> <span m="662980">all</span> <span m="663210">the</span> <span m="663270">way</span> <span m="663510">up</span> <span m="664230">to</span> <span m="664410">vk</span> <span m="665080">which</span> <span m="665270">is</span> <span m="665390">equal</span> <span m="665620">to</span> <span m="665760">v.</span> </p>
<p><span m="666480">So</span> <span m="666650">essentially,</span> <span m="668030">we</span> <span m="668140">have</span> <span m="668280">created</span> <span m="668660">a</span> <span m="668700">shorter</span> <span m="669100">walk.</span> <span m="669420">We have</span> <span m="669590">taken</span> <span m="669960">out</span> <span m="670130">this</span> <span m="670400">little</span> <span m="670630">triangle</span> <span m="671190">over</span> <span m="671460">here.</span> </p>
<p></p>
<p><span m="677260">Now</span> <span m="677440">this</span> <span m="677660">is</span> <span m="677810">a</span> <span m="677860">shorter</span> <span m="678320">walk.</span> <span m="680920">And</span> <span m="682940">this</span> <span m="683200">contradicts</span> <span m="684900">our</span> <span m="686250">assumption</span> <span m="686710">of</span> <span m="686800">minimality.</span> </p>
<p></p>
<p><span m="690250">So</span> <span m="690920">that</span> <span m="691100">means</span> <span m="691310">that</span> <span m="691620">our</span> <span m="692080">assumption</span> <span m="692650">over</span> <span m="692930">here</span> <span m="693650">is</span> <span m="693840">not</span> <span m="694010">true.</span> <span m="695740">So</span> <span m="695790">the</span> <span m="695890">walk</span> <span m="696300">is</span> <span m="696400">actually</span> <span m="696690">a</span> <span m="696980">path.</span> <span m="698350">So</span> <span m="698650">this</span> <span m="699040">concludes the</span> <span m="699530">proof.</span> </p>
<p></p>
<p><span m="704470">So</span> <span m="704680">even</span> <span m="704970">though</span> <span m="705130">this</span> <span m="705280">lemma</span> <span m="705700">is</span> <span m="706060">pretty</span> <span m="706300">straightforward</span> <span m="707050">in</span> <span m="707130">the</span> <span m="707230">sense</span> <span m="707600">that</span> <span m="707700">we</span> <span m="707840">all</span> <span m="708010">feel</span> <span m="708660">that</span> <span m="708880">if</span> <span m="709010">you</span> <span m="709140">have</span> <span m="709350">a walk,</span> <span m="709870">then</span> <span m="710080">you</span> <span m="710230">can</span> <span m="710400">create</span> <span m="710740">a</span> <span m="710820">path,</span> <span m="712010">if</span> <span m="712160">you</span> <span m="712280">really</span> <span m="712470">want</span> <span m="712640">to</span> <span m="712750">write</span> <span m="712970">it</span> <span m="713070">down</span> <span m="713720">with</span> <span m="713870">all</span> <span m="713970">the</span> <span m="714060">principles</span> <span m="714750">that</span> <span m="714820">we</span> <span m="714930">have</span> <span m="715190">been</span> <span m="715490">taught,</span> <span m="716870">you</span> <span m="717230">can</span> <span m="717360">see</span> <span m="717600">a</span> <span m="717640">few</span> <span m="717860">things</span> <span m="718160">appearing</span> <span m="718580">here.</span> <span m="718760">So</span> <span m="718910">we</span> <span m="719060">first</span> <span m="719360">started</span> <span m="719560">with the well</span> <span m="719890">ordering</span> <span m="720200">principle.</span> <span m="721170">And</span> <span m="721470">also</span> <span m="721800">we</span> <span m="721940">used</span> <span m="723470">the</span> <span m="723560">method</span> <span m="723890">of</span> <span m="724190">contradiction</span> <span m="724940">in</span> <span m="725080">our</span> <span m="725290">proof.</span> </p>
<p><span m="727620">So</span> <span m="727780">let's</span> <span m="728090">set</span> <span m="728520">talk</span> <span m="728740">about</span> <span m="729000">the</span> <span m="729070">next</span> <span m="729320">definition.</span> <span m="729910">So</span> <span m="730050">we</span> <span m="730150">talked</span> <span m="730410">about</span> <span m="730690">walks</span> <span m="731020">and</span> <span m="731150">paths.</span> <span m="732470">Let's</span> <span m="732630">talk</span> <span m="732810">about</span> <span m="733050">connectivity.</span> </p>
<p></p>
<p><span m="741460">We</span> <span m="741560">actually</span> <span m="741860">defined</span> <span m="743850">two</span> <span m="744130">vertices,</span> <span m="744580">u</span> <span m="744970">and</span> <span m="745100">v</span> <span m="745430">to</span> <span m="745540">be</span> <span m="745710">connected</span> <span m="747850">if</span> <span m="748110">there</span> <span m="748250">is</span> <span m="748370">a</span> <span m="748450">path</span> <span m="749410">that</span> <span m="749710">connects</span> <span m="750180">u</span> <span m="750480">to</span> <span m="750750">v.</span> <span m="751180">So</span> <span m="751570">u</span> <span m="751810">and v</span> <span m="752040">are</span> <span m="752200">connected</span> <span m="759160">if</span> <span m="759400">there</span> <span m="759570">is</span> <span m="759710">a</span> <span m="759760">path</span> <span m="764410">from</span> <span m="766540">u</span> <span m="767210">to</span> <span m="768110">v.</span> </p>
<p><span m="769680">And</span> <span m="771510">now</span> <span m="771770">we</span> <span m="771900">can</span> <span m="772030">talk</span> <span m="772430">about</span> <span m="772860">the</span> <span m="772930">graphs</span> <span m="773410">that</span> <span m="773480">are</span> <span m="773650">connected.</span> <span m="774540">That's</span> <span m="774740">a</span> <span m="774860">very</span> <span m="775080">important</span> <span m="775440">concept.</span> <span m="777090">A</span> <span m="777190">graph</span> <span m="777510">is</span> <span m="777680">connected</span> <span m="779370">if</span> <span m="789370">every</span> <span m="789510">pair</span> <span m="789790">of</span> <span m="789900">vertices</span> <span m="791540">is</span> <span m="791730">connected.</span> <span m="794480">So</span> <span m="794600">when</span> <span m="794840">every</span> <span m="795320">pair</span> <span m="796592">of</span> <span m="797020">vertices</span> <span m="797620">in</span> <span m="797780">the</span> <span m="797870">graph</span> <span m="804000">are</span> <span m="804210">connected.</span> </p>
<p><span m="805490">So</span> <span m="805780">an</span> <span m="805850">example</span> <span m="806150">is</span> <span m="806450">the</span> <span m="806720">particular</span> <span m="807130">graph</span> <span m="809710">on</span> <span m="810120">the</span> <span m="810220">blackboard.</span> <span m="812130">You</span> <span m="812280">can</span> <span m="812460">see</span> <span m="812790">here</span> <span m="813090">that</span> <span m="813445">there's</span> <span m="813800">always</span> <span m="814190">a</span> <span m="814260">path</span> <span m="814720">to</span> <span m="814830">be</span> <span m="814970">found</span> <span m="815350">from</span> <span m="815540">this</span> <span m="815790">vertex,</span> <span m="816440">to</span> <span m="816600">this</span> <span m="816880">one,</span> <span m="817110">or</span> <span m="817160">to</span> <span m="817290">that</span> <span m="817540">one,</span> <span m="818210">to</span> <span m="818330">this.</span> <span m="818530">to</span> <span m="818870">that,</span> <span m="819210">to</span> <span m="819330">this,</span> <span m="819620">or</span> <span m="819740">that.</span> <span m="822140">You</span> <span m="822290">can</span> <span m="823590">do</span> <span m="823720">this</span> <span m="823930">for</span> <span m="824120">any</span> <span m="824380">two</span> <span m="824600">vertices.</span> <span m="825420">So</span> <span m="825570">this</span> <span m="825750">is</span> <span m="825880">a</span> <span m="825950">connected</span> <span m="826330">graph.</span> </p>
<p><span m="827130">A graph</span> <span m="827470">that is</span> <span m="827730">not</span> <span m="827930">connected</span> <span m="829790">looks</span> <span m="830100">something</span> <span m="830500">like</span> <span m="833670">this</span> <span m="833890">for</span> <span m="834030">example.</span> <span m="834560">You</span> <span m="834610">may</span> <span m="834800">have</span> <span m="835190">a</span> <span m="835280">single</span> <span m="835710">edge</span> <span m="837930">plus,</span> <span m="838670">say,</span> <span m="838980">another</span> <span m="839270">triangle,</span> <span m="841770">maybe</span> <span m="842060">with</span> <span m="842240">another</span> <span m="842470">edge</span> <span m="842840">or</span> <span m="842920">something</span> <span m="843160">like</span> <span m="843320">that.</span> <span m="843490">So</span> <span m="843600">they</span> <span m="843770">are</span> <span m="844470">disconnected</span> <span m="845310">from</span> <span m="845520">this</span> <span m="845750">vertex.</span> <span m="846780">I</span> <span m="846900">do</span> <span m="847030">not</span> <span m="847330">have</span> <span m="847530">a</span> <span m="847600">graph</span> <span m="848020">that</span> <span m="848150">connects</span> <span m="848690">to</span> <span m="849000">this</span> <span m="849080">vertex</span> <span m="849490">over</span> <span m="849750">here.</span> </p>
<p><span m="852080">So</span> <span m="853100">this</span> <span m="853280">is</span> <span m="853750">about</span> <span m="854170">connectivity.</span> <span m="855640">And</span> <span m="857450">now</span> <span m="857750">we</span> <span m="857880">can</span> <span m="858360">start</span> <span m="858740">talking</span> <span m="859270">about</span> <span m="861350">cycles</span> <span m="861870">and</span> <span m="862070">closed</span> <span m="862430">walks.</span> </p>
<p></p>
<p><span m="870170">Together,</span> <span m="870650">these</span> <span m="870830">concepts</span> <span m="871490">will</span> <span m="873370">help</span> <span m="873660">us</span> <span m="873790">to</span> <span m="873900">define</span> <span m="874320">trees.</span> <span m="874960">And</span> <span m="875080">then</span> <span m="875190">we</span> <span m="875230">can</span> <span m="875390">start</span> <span m="875680">talking</span> <span m="875990">about</span> <span m="876210">this</span> <span m="876380">very</span> <span m="876560">special</span> <span m="877020">concept,</span> <span m="877510">spanning</span> <span m="877890">trees.</span> </p>
<p><span m="880980">So</span> <span m="881220">let's</span> <span m="881450">see</span> <span m="881680">how</span> <span m="881840">that</span> <span m="882010">works.</span> </p>
<p></p>
<p><span m="885930">So</span> <span m="886110">cycles</span> <span m="886940">and</span> <span m="887820">closed</span> <span m="888200">walks.</span> </p>
<p></p>
<p><span m="895110">We</span> <span m="895210">first</span> <span m="895480">define</span> <span m="896510">a</span> <span m="896650">closed</span> <span m="896990">walk.</span> <span m="898980">And</span> <span m="900270">once</span> <span m="900460">we</span> <span m="900530">have</span> <span m="900660">done</span> <span m="900840">this,</span> <span m="901790">we are</span> <span m="902020">able</span> <span m="902270">to</span> <span m="902390">get</span> <span m="902890">into</span> <span m="903150">cycles.</span> </p>
<p><span m="904210">So</span> <span m="904330">what's</span> <span m="904510">a</span> <span m="904580">closed</span> <span m="904910">walk?</span> <span m="906120">Well,</span> <span m="906670">it's</span> <span m="906860">actually a</span> <span m="907170">walk</span> <span m="907640">for</span> <span m="907780">which</span> <span m="907970">the</span> <span m="908060">start</span> <span m="908410">and</span> <span m="908510">the</span> <span m="908620">end</span> <span m="908830">vertex</span> <span m="909260">are</span> <span m="909380">the</span> <span m="909470">same.</span> </p>
<p></p>
<p><span m="915840">So</span> <span m="916180">it</span> <span m="916270">starts</span> <span m="918780">and</span> <span m="919030">ends</span> <span m="922650">at</span> <span m="922940">exactly</span> <span m="923390">the</span> <span m="923480">same</span> <span m="923730">vertex.</span> </p>
<p></p>
<p><span m="931770">So</span> <span m="931830">this</span> <span m="932000">is</span> <span m="932100">all</span> <span m="932330">pretty</span> <span m="932950">straightforward.</span> <span m="933830">So</span> <span m="933950">we</span> <span m="934070">have</span> <span m="934270">v0</span> <span m="934950">connected</span> <span m="935380">to,</span> <span m="935490">say,</span> <span m="935730">v1,</span> <span m="936430">all</span> <span m="936650">the</span> <span m="936730">way</span> <span m="936940">up</span> <span m="937110">to</span> <span m="937850">vk.</span> <span m="939360">And</span> <span m="939690">vk</span> <span m="940250">itself</span> <span m="941010">is</span> <span m="941200">equal</span> <span m="941470">to</span> <span m="941580">v0.</span> <span m="942650">So</span> <span m="942890">you</span> <span m="943050">have</span> <span m="943470">a</span> <span m="943580">walk</span> <span m="943880">that</span> <span m="944070">goes</span> <span m="944330">all</span> <span m="944550">the</span> <span m="944620">way</span> <span m="944790">from</span> <span m="944980">v0</span> <span m="945590">all</span> <span m="945830">the</span> <span m="945920">way</span> <span m="946070">back</span> <span m="946380">to</span> <span m="946500">vk.</span> </p>
<p><span m="947420">For</span> <span m="947540">example,</span> <span m="947840">in this</span> <span m="948010">graph,</span> <span m="949050">you</span> <span m="949160">could</span> <span m="949360">have</span> <span m="952100">walked</span> <span m="952440">all</span> <span m="952730">the</span> <span m="952830">way</span> <span m="953020">back</span> <span m="954110">to</span> <span m="954270">v0.</span> <span m="954920">You</span> <span m="955040">would</span> <span m="955150">have a</span> <span m="955390">closed</span> <span m="955670">walk.</span> </p>
<p><span m="956820">So</span> <span m="957080">what's</span> <span m="957210">a</span> <span m="957270">cycle?</span> <span m="958160">A</span> <span m="958260">cycle</span> <span m="958640">is</span> <span m="958770">a</span> <span m="958840">little</span> <span m="958980">bit</span> <span m="959120">more</span> <span m="959300">special</span> <span m="961000">in</span> <span m="961200">which</span> <span m="961660">all</span> <span m="961930">these</span> <span m="962180">vertices</span> <span m="962780">over</span> <span m="963080">here</span> <span m="963900">are</span> <span m="964050">actually</span> <span m="964370">different</span> <span m="964710">from</span> <span m="964860">one</span> <span m="965010">another.</span> <span m="966600">So</span> <span m="966970">if</span> <span m="968470">k</span> <span m="968500">is</span> <span m="968980">at</span> <span m="969080">least</span> <span m="969380">three--</span> <span m="969950">so</span> <span m="970880">the</span> <span m="970990">walk</span> <span m="971240">is</span> <span m="971370">not</span> <span m="971530">just</span> <span m="971810">a</span> <span m="971850">simple</span> <span m="972240">edge,</span> <span m="972990">or</span> <span m="974150">just a</span> <span m="974460">single</span> <span m="974780">vertex.</span> <span m="977790">And</span> <span m="977990">if</span> <span m="978310">all</span> <span m="978700">v0,</span> <span m="980230">v1,</span> <span m="981460">all</span> <span m="981680">the</span> <span m="981770">way</span> <span m="981930">up</span> <span m="982120">to</span> <span m="982180">vk</span> <span m="982730">minus</span> <span m="983170">1</span> <span m="984210">are</span> <span m="985120">different,</span> <span m="989340">then</span> <span m="990050">we</span> <span m="990320">are</span> <span m="992770">saying</span> <span m="995470">that</span> <span m="995680">we</span> <span m="995760">have</span> <span m="995940">a</span> <span m="996040">cycle.</span> <span m="996900">Then</span> <span m="997050">it</span> <span m="997180">is</span> <span m="997330">called</span> <span m="998490">a</span> <span m="998610">cycle.</span> </p>
<p></p>
<p><span m="1005250">So</span> <span m="1005440">these</span> <span m="1005720">two</span> <span m="1005880">concepts,</span> <span m="1006340">connectivity</span> <span m="1007170">and</span> <span m="1007320">cycles,</span> <span m="1008240">will</span> <span m="1008350">help</span> <span m="1008600">us</span> <span m="1008780">to</span> <span m="1008880">define</span> <span m="1009170">trees.</span> </p>
<p></p>
<p><span m="1015410">So</span> <span m="1015640">what</span> <span m="1015810">are</span> <span m="1015880">trees?</span> <span m="1017630">A simple</span> <span m="1017970">example</span> <span m="1018670">is</span> <span m="1018760">something</span> <span m="1019410">that</span> <span m="1019580">looks</span> <span m="1019910">like</span> <span m="1021590">this,</span> <span m="1022170">for</span> <span m="1022210">example.</span> </p>
<p></p>
<p><span m="1029980">The</span> <span m="1030119">idea</span> <span m="1030540">is that</span> <span m="1030795">a</span> <span m="1031050">tree</span> <span m="1031329">is</span> <span m="1031609">both</span> <span m="1031960">connected.</span> <span m="1033530">And</span> <span m="1033670">it</span> <span m="1033750">has</span> <span m="1033910">no</span> <span m="1034119">cycles.</span> <span m="1035630">So</span> <span m="1035720">in</span> <span m="1035770">this</span> <span m="1036010">case,</span> <span m="1036280">we</span> <span m="1036410">can</span> <span m="1036579">see</span> <span m="1036880">that</span> <span m="1038504">every</span> <span m="1038829">vertex</span> <span m="1039450">it's</span> <span m="1039640">connected</span> <span m="1039760">to</span> <span m="1039839">every</span> <span m="1040150">other</span> <span m="1040420">vertex.</span> <span m="1041310">But</span> <span m="1041420">there is</span> <span m="1041640">no</span> <span m="1041810">cycle</span> <span m="1042349">like</span> <span m="1042550">a</span> <span m="1042630">triangle</span> <span m="1043480">or</span> <span m="1043940">something</span> <span m="1044359">that</span> <span m="1044589">is connected</span> <span m="1045550">all</span> <span m="1045829">around.</span> </p>
<p><span m="1047220">So</span> <span m="1047510">let's</span> <span m="1047940">define</span> <span m="1048310">this</span> <span m="1049080">properly.</span> <span m="1050920">A</span> <span m="1051000">definition</span> <span m="1051610">is</span> <span m="1052410">that</span> <span m="1052710">a</span> <span m="1052890">connected</span> <span m="1057130">and</span> <span m="1057680">acyclic</span> <span m="1058380">graph</span> <span m="1063690">is</span> <span m="1063900">called</span> <span m="1064250">a</span> <span m="1064340">tree.</span> </p>
<p></p>
<p><span m="1071230">And</span> <span m="1071640">within</span> <span m="1072120">trees,</span> <span m="1072580">we</span> <span m="1072740">also</span> <span m="1073010">have</span> <span m="1073110">something</span> <span m="1073390">very</span> <span m="1073590">special</span> <span m="1073870">that</span> <span m="1074150">we call</span> <span m="1074560">leaves.</span> <span m="1075520">And</span> <span m="1075950">leaves</span> <span m="1076340">are</span> <span m="1076750">exactly</span> <span m="1076850">those</span> <span m="1077150">nodes</span> <span m="1078060">that</span> <span m="1078210">have</span> <span m="1079020">degree</span> <span m="1079370">one.</span> <span m="1079600">So</span> <span m="1079750">for</span> <span m="1079860">example,</span> <span m="1080200">this</span> <span m="1080410">node</span> <span m="1080610">has</span> <span m="1080800">degree</span> <span m="1081110">one.</span> <span m="1081740">It has</span> <span m="1081880">only</span> <span m="1082130">this</span> <span m="1082380">edge</span> <span m="1082700">connected</span> <span m="1083110">to</span> <span m="1083340">it.</span> <span m="1084290">So</span> <span m="1084600">this</span> <span m="1084790">is</span> <span m="1084940">called</span> <span m="1085210">a</span> <span m="1085480">leaf.</span> </p>
<p><span m="1086840">This</span> <span m="1087040">one</span> <span m="1087280">also</span> <span m="1087760">is a leaf</span> <span m="1088200">because</span> <span m="1088490">it has</span> <span m="1088760">just</span> <span m="1089600">one</span> <span m="1090710">edge</span> <span m="1091000">connected</span> <span m="1091360">to</span> <span m="1091490">it.</span> <span m="1091630">But</span> <span m="1091740">this</span> <span m="1091940">lead</span> <span m="1092270">over</span> <span m="1092400">here</span> <span m="1093030">has</span> <span m="1093340">three</span> <span m="1093700">edges.</span> <span m="1094290">So it</span> <span m="1094490">has</span> <span m="1094730">degree</span> <span m="1095060">three.</span> <span m="1096140">This</span> <span m="1096340">one</span> <span m="1096510">has</span> <span m="1096690">degree</span> <span m="1097030">four.</span> <span m="1097830">This</span> <span m="1098030">is</span> <span m="1098160">the</span> <span m="1098230">leaf</span> <span m="1098520">again.</span> <span m="1098850">This</span> <span m="1099070">one</span> <span m="1099270">is</span> <span m="1099560">a</span> <span m="1099850">leaf,</span> <span m="1100180">this</span> <span m="1100430">one,</span> <span m="1100790">and</span> <span m="1100960">this</span> <span m="1101170">one.</span> </p>
<p><span m="1103130">So</span> <span m="1103210">this</span> <span m="1103500">is</span> <span m="1103580">very</span> <span m="1103800">important</span> <span m="1104170">concept.</span> <span m="1105230">We</span> <span m="1105630">often</span> <span m="1106280">use</span> <span m="1106590">leaves</span> <span m="1106925">in</span> <span m="1107260">our</span> <span m="1107450">proofs.</span> <span m="1107920">And</span> <span m="1108120">we  will</span> <span m="1108220">get to</span> <span m="1108670">one.</span> </p>
<p><span m="1110510">So</span> <span m="1110890">a</span> <span m="1110950">lead</span> <span m="1115030">is</span> <span m="1115270">a</span> <span m="1115320">node</span> <span m="1118480">with</span> <span m="1119150">degree</span> <span m="1119520">one</span> <span m="1125310">in a tree.</span> </p>
<p></p>
<p><span m="1129810">OK</span> <span m="1130170">great.</span> </p>
<p></p>
<p><span m="1134020">So</span> <span m="1134200">now</span> <span m="1134340">let's</span> <span m="1134530">have</span> <span m="1134660">a</span> <span m="1134740">look</span> <span m="1134930">at</span> <span m="1135270">trees.</span> <span m="1136470">Suppose</span> <span m="1137370">I'm</span> <span m="1137550">looking</span> <span m="1137840">at</span> <span m="1137940">a</span> <span m="1137990">subgraph</span> <span m="1139500">of</span> <span m="1139690">a</span> <span m="1139800">tree.</span> <span m="1140410">What</span> <span m="1140560">can</span> <span m="1140680">we</span> <span m="1140780">say</span> <span m="1141040">about</span> <span m="1141240">a subgraph?</span> <span m="1143130">What</span> <span m="1143280">kind</span> <span m="1143420">of</span> <span m="1143480">structure</span> <span m="1143830">does</span> <span m="1144020">this</span> <span m="1144210">have?</span> </p>
<p><span m="1144530">So</span> <span m="1144900">suppose</span> <span m="1145370">for</span> <span m="1145490">example</span> <span m="1146450">I</span> <span m="1146630">want</span> <span m="1146940">to</span> <span m="1147170">take</span> <span m="1148400">some</span> <span m="1149180">connected</span> <span m="1149710">subgraphs.</span> <span m="1150380">So</span> <span m="1150490">that's</span> <span m="1150680">what</span> <span m="1150810">I'm</span> <span m="1150970">actually</span> <span m="1151310">interested</span> <span m="1151880">in.</span> <span m="1152060">So</span> <span m="1152120">for</span> <span m="1152230">example,</span> <span m="1152620">I</span> <span m="1152710">take</span> <span m="1153190">this--</span> <span m="1155080">suppose</span> <span m="1155400">I</span> <span m="1155500">take</span> <span m="1155730">this</span> <span m="1155930">connected</span> <span m="1156450">subgraph.</span> </p>
<p><span m="1158170">Then</span> <span m="1159440">what</span> <span m="1159670">does</span> <span m="1159850">it</span> <span m="1159970">look</span> <span m="1160140">like?</span> <span m="1161150">Do</span> <span m="1161250">we</span> <span m="1161260">see</span> <span m="1161470">some</span> <span m="1162510">structure</span> <span m="1162960">in</span> <span m="1163070">this?</span> <span m="1167890">It's--</span> <span m="1168790">no?</span> </p>
<p><span m="1169420">AUDIENCE: It's a</span> <span m="1169820">tree.</span> </p>
<p><span m="1170620">PROFESSOR: Yeah,</span> <span m="1170890">it's</span> <span m="1171090">a</span> <span m="1171120">three.</span> <span m="1171990">So</span> <span m="1173400">are</span> <span m="1173550">we</span> <span m="1173690">able</span> <span m="1173840">to</span> <span m="1173960">prove</span> <span m="1174280">this</span> <span m="1174440">also?</span> </p>
<p></p>
<p><span m="1177140">So</span> <span m="1177320">this</span> <span m="1177480">lemma</span> <span m="1177920">you</span> <span m="1178160">actually</span> <span m="1178520">lose</span> <span m="1178780">later</span> <span m="1179360">on.</span> <span m="1182060">So</span> <span m="1182450">any</span> <span m="1182820">connected</span> <span m="1184530">subgraph</span> <span m="1191050">of</span> <span m="1191250">the</span> <span m="1191370">tree</span> <span m="1194300">is</span> <span m="1194540">a</span> <span m="1194650">tree.</span> </p>
<p></p>
<p><span m="1197770">And</span> <span m="1198350">to</span> <span m="1198450">prove--</span> <span m="1199480">well,</span> <span m="1199890">how</span> <span m="1200070">can</span> <span m="1200210">we</span> <span m="1200300">start</span> <span m="1200630">out</span> <span m="1200780">a</span> <span m="1200880">proof?</span> <span m="1202440">Any</span> <span m="1202890">suggestions?</span> </p>
<p><span m="1204750">So</span> <span m="1204910">let's</span> <span m="1205070">have</span> <span m="1205240">a</span> <span m="1205310">look</span> <span m="1205510">here.</span> <span m="1207280">So</span> <span m="1207360">how</span> <span m="1207530">may</span> <span m="1207780">I</span> <span m="1207930">think</span> <span m="1208240">about</span> <span m="1208570">this?</span> <span m="1208960">Well,</span> <span m="1209250">I</span> <span m="1209340">may</span> <span m="1210090">take</span> <span m="1210640">a</span> <span m="1210790">connected</span> <span m="1211220">subgraph,</span> <span m="1212300">and</span> <span m="1212450">I</span> <span m="1212470">want</span> <span m="1212650">to</span> <span m="1212730">show--</span> <span m="1213640">of</span> <span m="1214010">a</span> <span m="1214100">tree-</span> <span m="1214530">and I want</span> <span m="1214630">show</span> <span m="1214850">that</span> <span m="1214950">it is</span> <span m="1215230">a tree.</span> </p>
<p><span m="1216880">A</span> <span m="1216950">general</span> <span m="1217280">method</span> <span m="1217650">could</span> <span m="1217870">be,</span> <span m="1218160">for</span> <span m="1218170">example,</span> <span m="1218660">to--</span> </p>
<p><span m="1221144">AUDIENCE: [INAUDIBLE]</span> <span m="1223088">You know,</span> <span m="1223574">one of the</span> <span m="1224060">things,</span> <span m="1224546">one of the</span> <span m="1225032">conditions</span> <span m="1225518">is</span> <span m="1226004">it</span> <span m="1226490">has</span> <span m="1226976">to be</span> <span m="1227462">acyclical.</span> <span m="1228434">So</span> <span m="1229406">since</span> <span m="1229892">you're</span> <span m="1230378">not adding</span> <span m="1230864">nodes and</span> <span m="1231350">you're not</span> <span m="1231836">starting</span> <span m="1232322">within</span> <span m="1232808">cycles,</span> <span m="1233294">you can't</span> <span m="1233780">create a</span> <span m="1234266">new</span> <span m="1234752">cycle.</span> </p>
<p><span m="1236210">PROFESSOR: Yeah.</span> <span m="1236460">I</span> <span m="1236510">think</span> <span m="1236760">you're</span> <span m="1236840">saying</span> <span m="1237130">something</span> <span m="1237530">like,</span> <span m="1238840">if</span> <span m="1239090">I</span> <span m="1239430">have a</span> <span m="1239670">connected</span> <span m="1239970">subgraph,</span> <span m="1241240">which</span> <span m="1241480">is</span> <span m="1241670">like</span> <span m="1242620">a</span> <span m="1242710">smaller</span> <span m="1243140">part</span> <span m="1243460">of</span> <span m="1243780">the</span> <span m="1243930">tree--</span> <span m="1245130">now</span> <span m="1245250">suppose</span> <span m="1245720">that</span> <span m="1245840">would</span> <span m="1245960">not</span> <span m="1246150">be</span> <span m="1246310">a tree.</span> <span m="1246520">Say it</span> <span m="1246790">would</span> <span m="1246900">have</span> <span m="1247050">a</span> <span m="1247090">cycle.</span> <span m="1248450">Then</span> <span m="1248810">that</span> <span m="1248980">cycle</span> <span m="1249330">would</span> <span m="1249430">also</span> <span m="1249670">be</span> <span m="1249800">present</span> <span m="1250160">in</span> <span m="1250250">the</span> <span m="1250350">tree</span> <span m="1250570">itself.</span> <span m="1251840">And</span> <span m="1251900">that's</span> <span m="1252070">not</span> <span m="1252220">really</span> <span m="1252430">possible</span> <span m="1252900">because</span> <span m="1253570">a</span> <span m="1253700">tree</span> <span m="1253870">has</span> <span m="1254100">no</span> <span m="1254230">cycles.</span> <span m="1256020">So</span> <span m="1256160">that's</span> <span m="1256400">sort</span> <span m="1256610">of</span> <span m="1256770">indeed</span> <span m="1257080">how</span> <span m="1257190">the</span> <span m="1257280">proof</span> <span m="1257550">goes.</span> <span m="1259440">So</span> <span m="1260110">we</span> <span m="1260170">essentially</span> <span m="1260700">use</span> <span m="1262170">contradiction.</span> </p>
<p></p>
<p><span m="1267630">So</span> <span m="1267890">let's</span> <span m="1268430">suppose</span> <span m="1271770">that this</span> <span m="1271940">connected</span> <span m="1272450">subgraph</span> <span m="1273310">is</span> <span m="1273520">actually</span> <span m="1273830">not</span> <span m="1274020">a</span> <span m="1274080">tree.</span> </p>
<p></p>
<p><span m="1284160">So</span> <span m="1284450">suppose</span> <span m="1284880">it's</span> <span m="1285020">a</span> <span m="1285080">connected</span> <span m="1285480">subgraph.</span> <span m="1285980">It's not a</span> <span m="1286100">tree.</span> <span m="1288040">Well</span> <span m="1288330">then,</span> <span m="1288750">in that</span> <span m="1289080">case,</span> <span m="1290410">it</span> <span m="1290570">must</span> <span m="1290790">have</span> <span m="1290910">a</span> <span m="1290970">cycle,</span> <span m="1291690">because</span> <span m="1292010">a</span> <span m="1292150">tree</span> <span m="1293150">is</span> <span m="1293390">something</span> <span m="1293830">that</span> <span m="1294030">is</span> <span m="1294220">both</span> <span m="1294510">connected</span> <span m="1295110">and</span> <span m="1295290">has</span> <span m="1295510">a</span> <span m="1295560">cycle</span> <span m="1296380">and</span> <span m="1297030">is</span> <span m="1297200">acyclic.</span> <span m="1298850">We</span> <span m="1299090">have</span> <span m="1299410">a</span> <span m="1299510">connected</span> <span m="1299900">subgraph,</span> <span m="1300850">which</span> <span m="1301050">is</span> <span m="1301170">not</span> <span m="1301350">a</span> <span m="1301410">tree.</span> <span m="1301590">So</span> <span m="1301990">it</span> <span m="1302200">must have</span> <span m="1302360">a</span> <span m="1302410">cycle.</span> </p>
<p><span m="1305380">So</span> <span m="1305590">it</span> <span m="1305690">has</span> <span m="1305920">a</span> <span m="1306030">cycle.</span> <span m="1308480">But</span> <span m="1308660">now</span> <span m="1308830">since</span> <span m="1309110">it</span> <span m="1309200">is</span> <span m="1309340">a</span> <span m="1309560">subgraph</span> <span m="1310140">of</span> <span m="1310330">the</span> <span m="1310410">bigger</span> <span m="1310690">graph,</span> <span m="1312400">we</span> <span m="1312520">know</span> <span m="1312790">that</span> <span m="1312930">the</span> <span m="1312970">cycle</span> <span m="1313790">must</span> <span m="1313940">also</span> <span m="1314210">be in</span> <span m="1314420">the</span> <span m="1314510">whole</span> <span m="1314750">graph.</span> <span m="1315780">So</span> <span m="1315990">the</span> <span m="1316090">whole</span> <span m="1316380">graph</span> <span m="1322740">has</span> <span m="1324476">this</span> <span m="1324910">particular</span> <span m="1325370">cycle.</span> </p>
<p><span m="1327190">Wait</span> <span m="1327380">a</span> <span m="1327410">minute.</span> <span m="1329620">The</span> <span m="1329760">whole</span> <span m="1329980">graph</span> <span m="1330410">is</span> <span m="1330590">the</span> <span m="1330720">tree.</span> <span m="1333420">And</span> <span m="1333600">a</span> <span m="1333700">tree's</span> <span m="1334000">acyclic.</span> <span m="1335130">So</span> <span m="1335350">we</span> <span m="1335450">get</span> <span m="1335610">a</span> <span m="1335680">contradiction.</span> </p>
<p><span m="1337280">So</span> <span m="1337380">that</span> <span m="1337480">means</span> <span m="1337770">that</span> <span m="1337850">our</span> <span m="1338160">original</span> <span m="1338570">assumption</span> <span m="1339800">that</span> <span m="1339970">it's not a</span> <span m="1340100">tree</span> <span m="1340610">is</span> <span m="1341450">wrong.</span> <span m="1341890">So</span> <span m="1342060">it</span> <span m="1342150">is</span> <span m="1342330">a</span> <span m="1342410">tree.</span> <span m="1343580">So</span> <span m="1343730">this</span> <span m="1343940">is,</span> <span m="1344740">again,</span> <span m="1345840">such</span> <span m="1346110">a</span> <span m="1346160">general</span> <span m="1346760">kind</span> <span m="1346990">of</span> <span m="1347110">proof</span> <span m="1347390">method</span> <span m="1347725">that</span> <span m="1348060">we</span> <span m="1348200">repeatedly</span> <span m="1348690">use</span> <span m="1349000">here.</span> </p>
<p><span m="1350520">So</span> <span m="1351400">this</span> <span m="1351670">lemma,</span> <span m="1352410">which we</span> <span m="1352850">will</span> <span m="1353130">call</span> <span m="1353460">two,</span> <span m="1355380">we</span> <span m="1355500">will</span> <span m="1355630">use</span> <span m="1356160">in</span> <span m="1356290">the</span> <span m="1356380">next</span> <span m="1356720">proof</span> <span m="1357390">where</span> <span m="1357820">we're going to</span> <span m="1357950">talk</span> <span m="1358260">about</span> <span m="1359670">the</span> <span m="1359770">relationship</span> <span m="1360370">between</span> <span m="1360660">vertices</span> <span m="1360970">and</span> <span m="1361500">edges</span> <span m="1362000">within</span> <span m="1363720">trees.</span> </p>
<p></p>
<p><span m="1375980">It's</span> <span m="1376200">a</span> <span m="1376260">very</span> <span m="1377690">beautiful</span> <span m="1378050">relationship.</span> <span m="1379450">The</span> <span m="1379640">lemma</span> <span m="1380060">states</span> <span m="1382120">if</span> <span m="1382260">you</span> <span m="1382360">have</span> <span m="1382530">a</span> <span m="1382670">tree</span> <span m="1383110">that</span> <span m="1383240">has</span> <span m="1383550">n</span> <span m="1383840">vertices,</span> <span m="1385150">then</span> <span m="1385280">it</span> <span m="1385380">must</span> <span m="1385700">have</span> <span m="1385940">n</span> <span m="1386150">minus</span> <span m="1386520">1</span> <span m="1386710">edges.</span> <span m="1387840">So</span> <span m="1388000">this</span> <span m="1388210">is a</span> <span m="1388390">very</span> <span m="1388620">tight</span> <span m="1388920">relationship</span> <span m="1389510">here.</span> </p>
<p><span m="1391110">So</span> <span m="1391400">a</span> <span m="1391600">tree</span> <span m="1393130">with</span> <span m="1395180">n</span> <span m="1395440">vertices</span> <span m="1399490">has</span> <span m="1400110">n</span> <span m="1400300">minus</span> <span m="1400710">1</span> <span m="1401260">edges.</span> </p>
<p><span m="1403070">So</span> <span m="1403260">let's</span> <span m="1403510">see.</span> <span m="1403860">How</span> <span m="1404130">can</span> <span m="1404250">we</span> <span m="1404360">prove</span> <span m="1404600">this</span> <span m="1404800">one?</span> <span m="1406930">Any</span> <span m="1407150">suggestions?</span> </p>
<p><span m="1407770">So</span> <span m="1408010">what</span> <span m="1409030">did</span> <span m="1409380">we</span> <span m="1409530">use</span> <span m="1410400">at</span> <span m="1410540">the</span> <span m="1410610">beginning</span> <span m="1410980">of</span> <span m="1411120">the</span> <span m="1411220">course</span> <span m="1411570">here?</span> <span m="1412600">So</span> <span m="1412790">we</span> <span m="1412910">have</span> <span m="1413120">n</span> <span m="1413310">vertices.</span> <span m="1415040">And</span> <span m="1416080">we</span> <span m="1416200">want</span> <span m="1416390">to</span> <span m="1416470">show</span> <span m="1417390">that</span> <span m="1417690">it</span> <span m="1417840">has</span> <span m="1418000">n</span> <span m="1418190">minus</span> <span m="1418500">1</span> <span m="1418680">edges.</span> </p>
<p></p>
<p><span m="1422980">We</span> <span m="1423160">can</span> <span m="1423330">use</span> <span m="1423650">induction,</span> <span m="1424230">right?</span> <span m="1425670">So</span> <span m="1425920">let's</span> <span m="1426810">use</span> <span m="1427560">induction.</span> <span m="1430350">on</span> <span m="1430680">n.</span> </p>
<p><span m="1431720">So</span> <span m="1431890">how</span> <span m="1432030">do</span> <span m="1432150">we</span> <span m="1432380">proceed</span> <span m="1432740">if</span> <span m="1432900">you</span> <span m="1432970">do</span> <span m="1433150">this?</span> <span m="1434280">Well,</span> <span m="1434410">we</span> <span m="1434510">always</span> <span m="1434830">start</span> <span m="1435150">out</span> <span m="1435490">with</span> <span m="1437130">an</span> <span m="1437250">induction</span> <span m="1437900">hypothesis.</span> <span m="1438920">So</span> <span m="1439150">the</span> <span m="1439250">statement</span> <span m="1439700">here</span> <span m="1439950">would</span> <span m="1440080">be</span> <span m="1440190">something</span> <span m="1440640">like,</span> <span m="1442360">well,</span> <span m="1443950">there</span> <span m="1444250">are</span> <span m="1445880">n</span> <span m="1446450">minus</span> <span m="1446890">1</span> <span m="1447210">edges</span> <span m="1449600">in</span> <span m="1450100">any</span> <span m="1451870">n</span> <span m="1452170">vertex</span> <span m="1455040">tree.</span> </p>
<p></p>
<p><span m="1458480">And</span> <span m="1459530">if</span> <span m="1459700">you</span> <span m="1459780">start</span> <span m="1460050">with</span> <span m="1460170">induction,</span> <span m="1460630">you always</span> <span m="1460950">start</span> <span m="1461180">with</span> <span m="1461250">the</span> <span m="1461330">base</span> <span m="1461630">case,</span> <span m="1461880">right?</span> <span m="1462580">So</span> <span m="1462930">we</span> <span m="1463130">have the</span> <span m="1463200">base</span> <span m="1463540">case.</span> </p>
<p><span m="1465930">So</span> <span m="1466140">how</span> <span m="1466270">does</span> <span m="1466450">this</span> <span m="1466650">work?</span> <span m="1468540">So</span> <span m="1468770">for</span> <span m="1468980">example,</span> <span m="1470680">we</span> <span m="1470830">want</span> <span m="1470980">to</span> <span m="1471080">consider</span> <span m="1471490">P1.</span> <span m="1473090">So</span> <span m="1473850">there</span> <span m="1474080">are</span> <span m="1474330">zero</span> <span m="1474960">edges</span> <span m="1475580">in</span> <span m="1475810">any</span> <span m="1476150">one</span> <span m="1476520">vertex</span> <span m="1476890">tree.</span> <span m="1477260">Well,</span> <span m="1477380">that's</span> <span m="1477620">true,</span> <span m="1477910">right?</span> <span m="1478110">If  I</span> <span m="1478300">have just</span> <span m="1478540">one</span> <span m="1478730">vertex,</span> <span m="1479720">there's</span> <span m="1479920">no</span> <span m="1480330">edge.</span> <span m="1481390">So</span> <span m="1482790">this</span> <span m="1482980">is</span> <span m="1483090">definitely</span> <span m="1483500">correct.</span> </p>
<p><span m="1485630">So</span> <span m="1485800">that's</span> <span m="1486010">easy.</span> <span m="1486930">So</span> <span m="1487060">let's</span> <span m="1487300">prove</span> <span m="1487800">the</span> <span m="1488080">other</span> <span m="1488280">part,</span> <span m="1489300">which</span> <span m="1489570">is</span> <span m="1489740">the</span> <span m="1491470">inductive</span> <span m="1491920">step.</span> <span m="1493900">And</span> <span m="1494210">the</span> <span m="1494420">inductive</span> <span m="1494470">step</span> <span m="1498870">is--</span> </p>
<p></p>
<p><span m="1504770">So</span> <span m="1504960">to</span> <span m="1505090">do</span> <span m="1505200">the</span> <span m="1505320">inductive</span> <span m="1505810">step,</span> <span m="1507430">we are</span> <span m="1507590">going</span> <span m="1507850">to</span> <span m="1508030">always</span> <span m="1508340">assume</span> <span m="1510360">Pn.</span> <span m="1511270">And</span> <span m="1511390">then we</span> <span m="1511520">want</span> <span m="1511690">to</span> <span m="1511790">prove</span> <span m="1512140">Pn</span> <span m="1512510">plus</span> <span m="1512750">1.</span> </p>
<p><span m="1514530">So</span> <span m="1514760">let's</span> <span m="1514960">do</span> <span m="1515100">this.</span> <span m="1516060">So</span> <span m="1516180">we</span> <span m="1516300">have</span> <span m="1516440">the</span> <span m="1516530">inductive</span> <span m="1516940">step.</span> </p>
<p></p>
<p><span m="1522740">And</span> <span m="1522900">we</span> <span m="1523000">always</span> <span m="1523320">start</span> <span m="1523650">out</span> <span m="1523810">the</span> <span m="1523890">same</span> <span m="1524230">way.</span> <span m="1524780">So</span> <span m="1525080">we</span> <span m="1525190">suppose</span> <span m="1529610">P</span> <span m="1530980">of n.</span> <span m="1532850">And</span> <span m="1533160">now</span> <span m="1533320">we</span> <span m="1533380">want</span> <span m="1533610">to</span> <span m="1533690">prove</span> <span m="1534010">Pn</span> <span m="1534360">plus</span> <span m="1534600">1.</span> <span m="1534960">So</span> <span m="1536080">how</span> <span m="1536180">do</span> <span m="1536270">we</span> <span m="1536440">do</span> <span m="1536590">this?</span> </p>
<p><span m="1537350">Well,</span> <span m="1537550">take</span> <span m="1538430">a</span> <span m="1538610">tree</span> <span m="1538900">that</span> <span m="1539010">has</span> <span m="1539200">n</span> <span m="1539460">plus</span> <span m="1539580">1</span> <span m="1539760">vertices</span> <span m="1540200">vertices.</span> <span m="1540910">And we</span> <span m="1541080">want</span> <span m="1541200">to</span> <span m="1541290">show</span> <span m="1541770">that</span> <span m="1542045">it</span> <span m="1542320">has</span> <span m="1542770">n</span> <span m="1543090">edges.</span> <span m="1544390">So</span> <span m="1545800">take</span> <span m="1546120">any</span> <span m="1546370">such</span> <span m="1546690">tree.</span> <span m="1547680">So let</span> <span m="1548105">T</span> <span m="1549710">be</span> <span m="1549990">a</span> <span m="1550120">tree</span> <span m="1552020">that</span> <span m="1552500">has</span> <span m="1554420">n</span> <span m="1555380">plus</span> <span m="1555610">1</span> <span m="1556540">vertices.</span> </p>
<p></p>
<p><span m="1562170">So</span> <span m="1562770">how</span> <span m="1563210">can</span> <span m="1563390">we</span> <span m="1563500">use</span> <span m="1563800">the</span> <span m="1563920">induction</span> <span m="1564700">hypothesis</span> <span m="1565360">here?</span> <span m="1566880">So</span> <span m="1567470">do</span> <span m="1567870">you</span> <span m="1568020">have</span> <span m="1568190">any</span> <span m="1568400">ideas</span> <span m="1568760">over</span> <span m="1569140">here?</span> <span m="1569360">So</span> <span m="1569940">I</span> <span m="1570040">have</span> <span m="1570270">a tree</span> <span m="1570790">with</span> <span m="1570960">n</span> <span m="1571130">plus</span> <span m="1571370">1</span> <span m="1571530">vertices.</span> <span m="1573100">And</span> <span m="1573490">if</span> <span m="1573670">I want</span> <span m="1573830">to</span> <span m="1573950">be</span> <span m="1574060">able</span> <span m="1574270">to</span> <span m="1574390">use</span> <span m="1574630">this</span> <span m="1574790">induction</span> <span m="1576960">at</span> <span m="1577110">this</span> <span m="1577280">induction</span> <span m="1577720">hypothesis,</span> <span m="1578420">then</span> <span m="1579560">I</span> <span m="1579740">can</span> <span m="1579990">only</span> <span m="1580210">apply</span> <span m="1580680">it on</span> <span m="1580830">trees</span> <span m="1583080">that</span> <span m="1583220">have</span> <span m="1583500">n</span> <span m="1583780">vertices.</span> <span m="1584125">So</span> <span m="1584470">somehow</span> <span m="1584860">I</span> <span m="1585090">have to</span> <span m="1585170">delete</span> <span m="1585550">one</span> <span m="1585710">vertex.</span> <span m="1586960">Right?</span> </p>
<p><span m="1588440">So</span> <span m="1588670">what</span> <span m="1588890">kind</span> <span m="1589090">of</span> <span m="1589240">vertex</span> <span m="1589600">can</span> <span m="1589790">I</span> <span m="1589870">delete</span> <span m="1590760">in</span> <span m="1590850">such</span> <span m="1591220">a</span> <span m="1591320">way</span> <span m="1592120">that</span> <span m="1593360">we</span> <span m="1593470">still</span> <span m="1593900">have a</span> <span m="1594210">tree?</span> </p>
<p></p>
<p><span m="1598730">And</span> <span m="1598880">I</span> <span m="1598920">need</span> <span m="1599160">that</span> <span m="1599420">because</span> <span m="1600530">this</span> <span m="1602860">induction</span> <span m="1603350">step</span> <span m="1604240">can</span> <span m="1604370">only</span> <span m="1604550">be</span> <span m="1604670">applied</span> <span m="1605270">to</span> <span m="1605660">trees.</span> </p>
<p><span m="1607660">So</span> <span m="1608410">what</span> <span m="1608620">type</span> <span m="1608820">of</span> <span m="1608910">vertex</span> <span m="1609120">can</span> <span m="1609290">I</span> <span m="1609470">remove</span> <span m="1609780">from</span> <span m="1609930">a</span> <span m="1610060">tree</span> <span m="1610460">and</span> <span m="1610680">it still</span> <span m="1613060">stays</span> <span m="1613490">a tree?</span> <span m="1614560">So</span> <span m="1614750">for</span> <span m="1614890">example,</span> <span m="1615360">if</span> <span m="1615460">you</span> <span m="1615560">look</span> <span m="1615640">at</span> <span m="1616080">this</span> <span m="1616310">example</span> <span m="1616800">up</span> <span m="1616930">here,</span> <span m="1619630">what</span> <span m="1619810">kind</span> <span m="1619990">of</span> <span m="1620100">vertex</span> <span m="1620510">can</span> <span m="1620630">I</span> <span m="1620730">remove</span> <span m="1621070">here?</span> </p>
<p><span m="1621540">AUDIENCE: A</span> <span m="1621933">leaf.</span> </p>
<p><span m="1622326">PROFESSOR: Yeah,</span> <span m="1622720">except.</span> <span m="1624230">I</span> <span m="1624330">can</span> <span m="1624600">remove</span> <span m="1625090">a leaf--</span> <span m="1625210">for example,</span> <span m="1625640">this</span> <span m="1625910">one.</span> <span m="1626630">And</span> <span m="1626770">why</span> <span m="1626940">is</span> <span m="1627090">that?</span> <span m="1627260">Because</span> <span m="1627580">if there's</span> <span m="1627760">only</span> <span m="1628070">one</span> <span m="1628430">edge</span> <span m="1628870">connected</span> <span m="1629290">to</span> <span m="1629400">the</span> <span m="1629510">rest</span> <span m="1629790">of</span> <span m="1629910">the</span> <span m="1630020">tree--</span> <span m="1630220">so</span> <span m="1630350">if</span> <span m="1630490">I</span> <span m="1632400">delete</span> <span m="1632803">it,</span> <span m="1633610">I</span> <span m="1633760">will</span> <span m="1633970">actually</span> <span m="1634610">keep</span> <span m="1634900">a</span> <span m="1635050">tree.</span> <span m="1635420">So</span> <span m="1635510">that's</span> <span m="1635850">how</span> <span m="1636040">the</span> <span m="1636180">proof</span> <span m="1636500">will</span> <span m="1636610">continue.</span> </p>
<p><span m="1638070">So</span> <span m="1638150">we</span> <span m="1638290">take</span> <span m="1638530">out</span> <span m="1638700">one</span> <span m="1639090">vertex.</span> <span m="1641170">So</span> <span m="1642750">let</span> <span m="1642940">v</span> <span m="1643830">be</span> <span m="1644455">a</span> <span m="1644730">leaf</span> <span m="1645270">of the</span> <span m="1645480">tree.</span> </p>
<p></p>
<p><span m="1650920">And</span> <span m="1652220">let's</span> <span m="1652740">remove</span> <span m="1653850">this</span> <span m="1654060">particular</span> <span m="1655680">vertex.</span> </p>
<p></p>
<p><span m="1659230">So</span> <span m="1659380">what</span> <span m="1659530">do</span> <span m="1659610">we</span> <span m="1659750">get?</span> <span m="1661250">Well,</span> <span m="1661440">we</span> <span m="1661630">again</span> <span m="1662350">have</span> <span m="1662970">a</span> <span m="1665140">connected</span> <span m="1665650">subgraph.</span> <span m="1667600">So</span> <span m="1667860">this</span> <span m="1668110">creates</span> <span m="1670600">a</span> <span m="1670890">connected</span> <span m="1674270">subgraph.</span> <span m="1676540">And</span> <span m="1678660">we</span> <span m="1679220">also</span> <span m="1679570">know</span> <span m="1679800">that</span> <span m="1679930">this</span> <span m="1680110">is</span> <span m="1680240">a</span> <span m="1680360">tree.</span> <span m="1680740">So how</span> <span m="1681320">can we</span> <span m="1681780">conclude</span> <span m="1682020">that?</span> </p>
<p><span m="1682640">Well,</span> <span m="1682910">we</span> <span m="1683060">had</span> <span m="1683340">another</span> <span m="1683640">lemma</span> <span m="1683990">somewhere</span> <span m="1685060">that's</span> <span m="1685280">behind</span> <span m="1685850">this</span> <span m="1686940">blackboard.</span> <span m="1689060">It</span> <span m="1689210">says</span> <span m="1689640">that</span> <span m="1690040">if</span> <span m="1690210">you</span> <span m="1690310">have</span> <span m="1690550">connected</span> <span m="1690820">subgraph,</span> <span m="1691555">then</span> <span m="1691910">we</span> <span m="1692040">still</span> <span m="1692300">have</span> <span m="1692470">a</span> <span m="1692570">tree.</span> </p>
<p><span m="1693896">So</span> <span m="1694340">this is</span> <span m="1694620">great.</span> <span m="1695040">So</span> <span m="1695260">this</span> <span m="1695470">is</span> <span m="1695600">also</span> <span m="1696695">a</span> <span m="1699270">tree.</span> </p>
<p></p>
<p><span m="1702760">So</span> <span m="1702910">now</span> <span m="1703440">we</span> <span m="1703650">can</span> <span m="1703850">apply</span> <span m="1705730">Pn</span> <span m="1706620">because</span> <span m="1706960">we</span> <span m="1707070">have</span> <span m="1707350">n</span> <span m="1707710">vertices,</span> <span m="1708290">because</span> <span m="1708540">we</span> <span m="1708640">deleted</span> <span m="1709060">one.</span> <span m="1710140">So</span> <span m="1710400">by</span> <span m="1710670">Pn,</span> <span m="1713410">we</span> <span m="1713530">can</span> <span m="1713710">conclude</span> <span m="1714280">that</span> <span m="1714600">this</span> <span m="1714830">particular</span> <span m="1715450">subgraph</span> <span m="1716630">has</span> <span m="1717490">n</span> <span m="1717880">minus</span> <span m="1718280">1</span> <span m="1718680">edges.</span> <span m="1720510">You</span> <span m="1720600">simply</span> <span m="1721020">apply</span> <span m="1721480">our</span> <span m="1721710">statement</span> <span m="1722470">over</span> <span m="1722740">here.</span> </p>
<p><span m="1726560">But</span> <span m="1726690">we</span> <span m="1726800">want</span> <span m="1726930">to</span> <span m="1727020">say</span> <span m="1727200">something</span> <span m="1727520">about</span> <span m="1728020">the</span> <span m="1728505">original</span> <span m="1728990">tree.</span> <span m="1729960">So</span> <span m="1730100">how</span> <span m="1730220">do</span> <span m="1730310">we</span> <span m="1730410">do</span> <span m="1730570">this?</span> <span m="1731680">Well,</span> <span m="1731700">we</span> <span m="1731830">have</span> <span m="1732020">to</span> <span m="1732130">somehow</span> <span m="1732510">reconstruct</span> <span m="1732805">this</span> <span m="1733100">tree.</span> <span m="1733840">Well,</span> <span m="1733990">we</span> <span m="1734110">have</span> <span m="1734400">deleted</span> <span m="1734860">v.</span> <span m="1735600">So</span> <span m="1735780">we</span> <span m="1735880">have</span> <span m="1736040">to</span> <span m="1736150">reattach</span> <span m="1736555">v</span> <span m="1736960">again.</span> <span m="1739020">So</span> <span m="1739270">let's</span> <span m="1739490">do</span> <span m="1739640">this.</span> </p>
<p><span m="1742630">So</span> <span m="1742850">we</span> <span m="1743040">reattach</span> <span m="1745474">v.</span> <span m="1747390">And</span> <span m="1747570">if</span> <span m="1747700">I</span> <span m="1747770">do</span> <span m="1747980">this,</span> <span m="1748730">well,</span> <span m="1749210">let's</span> <span m="1749540">look</span> <span m="1749710">at</span> <span m="1749770">this</span> <span m="1749940">example</span> <span m="1750420">over</span> <span m="1750670">here.</span> <span m="1751310">I</span> <span m="1751460">reattach</span> <span m="1751775">the</span> <span m="1752090">leaf.</span> <span m="1752950">I</span> <span m="1753060">will</span> <span m="1753320">add</span> <span m="1753580">one</span> <span m="1753750">more</span> <span m="1753990">edge.</span> </p>
<p><span m="1756260">So</span> <span m="1756880">if</span> <span m="1757080">I</span> <span m="1757360">reattach</span> <span m="1758060">this</span> <span m="1758230">particular</span> <span m="1758950">leaf,</span> <span m="1761100">I</span> <span m="1761220">will</span> <span m="1761410">get</span> <span m="1761630">back</span> <span m="1761920">the</span> <span m="1762090">original</span> <span m="1764890">T</span> <span m="1765560">over</span> <span m="1765880">here.</span> </p>
<p><span m="1767100">And</span> <span m="1768850">how</span> <span m="1769000">many</span> <span m="1769470">edges</span> <span m="1769600">does</span> <span m="1769780">it</span> <span m="1769880">have?</span> <span m="1770600">Well,</span> <span m="1770950">it has</span> <span m="1771180">these</span> <span m="1771420">n</span> <span m="1771620">minus</span> <span m="1771950">1</span> <span m="1772270">edges</span> <span m="1775650">plus</span> <span m="1776430">the</span> <span m="1776640">one</span> <span m="1777040">edge</span> <span m="1777370">that</span> <span m="1777510">is</span> <span m="1777690">needed</span> <span m="1778090">to</span> <span m="1778360">reattach</span> <span m="1779030">the</span> <span m="1779530">leaf.</span> <span m="1781890">So</span> <span m="1782060">that's</span> <span m="1782420">all</span> <span m="1782640">together</span> <span m="1783510">exactly</span> <span m="1784200">n</span> <span m="1784780">edges.</span> </p>
<p><span m="1786420">So</span> <span m="1786620">now</span> <span m="1786910">we</span> <span m="1787010">have</span> <span m="1787640">shown</span> <span m="1788540">that</span> <span m="1788780">for</span> <span m="1789320">every</span> <span m="1789900">n</span> <span m="1790110">plus</span> <span m="1790350">1</span> <span m="1790540">vertex</span> <span m="1790810">tree,</span> <span m="1794430">every</span> <span m="1794590">such</span> <span m="1794935">tree</span> <span m="1795280">has</span> <span m="1795470">n</span> <span m="1795940">edges.</span> </p>
<p><span m="1797440">So</span> <span m="1797860">now</span> <span m="1798020">we're</span> <span m="1798160">done,</span> <span m="1798580">because</span> <span m="1798940">this</span> <span m="1799120">shows</span> <span m="1800240">P</span> <span m="1801030">n</span> <span m="1801250">plus</span> <span m="1801500">1.</span> <span m="1802740">So</span> <span m="1802880">this</span> <span m="1803060">is</span> <span m="1803170">how</span> <span m="1803320">the lemma</span> <span m="1804580">is</span> <span m="1804810">proved.</span> <span m="1805105">And</span> <span m="1805400">in</span> <span m="1805830">the</span> <span m="1805970">homework,</span> <span m="1806570">you</span> <span m="1806680">actually</span> <span m="1808290">use</span> <span m="1808610">this.</span> </p>
<p></p>
<p><span m="1812300">So</span> <span m="1813640">let's</span> <span m="1813900">now</span> <span m="1814060">talk</span> <span m="1814340">about</span> <span m="1814630">very</span> <span m="1814830">special</span> <span m="1816680">trees,</span> <span m="1817190">spanning</span> <span m="1817630">trees.</span> </p>
<p></p>
<p><span m="1820290">I</span> <span m="1820370">think</span> <span m="1820570">this</span> <span m="1820710">is</span> <span m="1820800">a</span> <span m="1820870">very</span> <span m="1821070">exciting</span> <span m="1821800">topic.</span> <span m="1822760">So</span> <span m="1823880">let's</span> <span m="1824500">look</span> <span m="1824750">at</span> <span m="1824840">this</span> <span m="1825000">particular</span> <span m="1826020">graph</span> <span m="1826420">over</span> <span m="1826700">here.</span> </p>
<p></p>
<p><span m="1833330">Let's</span> <span m="1833560">define</span> <span m="1834020">spanning</span> <span m="1834470">trees.</span> <span m="1835380">So</span> <span m="1835560">what's</span> <span m="1835760">a</span> <span m="1835820">spanning</span> <span m="1836230">tree?</span> <span m="1836640">A spanning</span> <span m="1837050">tree</span> <span m="1837310">is a</span> <span m="1837440">subgraph</span> <span m="1837825">of</span> <span m="1838640">a</span> <span m="1838890">graph</span> <span m="1840020">that</span> <span m="1840920">somehow</span> <span m="1841430">spans</span> <span m="1842190">all</span> <span m="1842420">the</span> <span m="1842530">vertices</span> <span m="1843180">within</span> <span m="1843520">this</span> <span m="1843750">graph.</span> <span m="1844680">So</span> <span m="1845440">it's</span> <span m="1845870">a</span> <span m="1846090">tree</span> <span m="1847560">that</span> <span m="1847800">touches</span> <span m="1848230">every</span> <span m="1848500">single</span> <span m="1848860">vertex.</span> </p>
<p><span m="1849830">So</span> <span m="1850040">for</span> <span m="1850220">example,</span> <span m="1852240">we</span> <span m="1852390">may</span> <span m="1852690">have--</span> </p>
<p></p>
<p><span m="1856030">Maybe</span> <span m="1856420">this is</span> <span m="1856680">not</span> <span m="1856800">such</span> <span m="1857060">a</span> <span m="1857110">bright</span> <span m="1857390">color.</span> <span m="1859690">Maybe</span> <span m="1860150">this</span> <span m="1860700">one.</span> </p>
<p><span m="1862460">So</span> <span m="1862660">for</span> <span m="1862820">example,</span> <span m="1866040">we</span> <span m="1866070">may</span> <span m="1866240">have</span> <span m="1866420">a</span> <span m="1866530">tree</span> <span m="1866790">that</span> <span m="1866920">looks</span> <span m="1867120">like</span> <span m="1867320">this.</span> </p>
<p></p>
<p><span m="1876210">So</span> <span m="1876460">in</span> <span m="1876610">this</span> <span m="1877540">particular</span> <span m="1878020">graph,</span> <span m="1879250">we</span> <span m="1879350">can</span> <span m="1879450">create</span> <span m="1879860">a</span> <span m="1879910">subgraph</span> <span m="1881830">with</span> <span m="1882190">the</span> <span m="1883030">thickened</span> <span m="1883580">green</span> <span m="1884430">edges</span> <span m="1884840">over</span> <span m="1885120">here,</span> <span m="1886280">which</span> <span m="1886370">is</span> <span m="1886520">a</span> <span m="1886670">tree,</span> <span m="1887160">and</span> <span m="1887430">also</span> <span m="1887830">covers</span> <span m="1888480">all</span> <span m="1889060">the</span> <span m="1889160">different</span> <span m="1889560">vertices</span> <span m="1890210">of</span> <span m="1890520">the</span> <span m="1890580">original</span> <span m="1890930">graph.</span> </p>
<p><span m="1893180">This is</span> <span m="1893360">actually</span> <span m="1893670">pretty</span> <span m="1893940">amazing</span> <span m="1894410">that</span> <span m="1894560">you</span> <span m="1894650">can</span> <span m="1894760">do</span> <span m="1894900">this,</span> <span m="1895180">you</span> <span m="1895330">can</span> <span m="1895550">actually</span> <span m="1896110">create</span> <span m="1896490">such</span> <span m="1897710">a</span> <span m="1897820">kind</span> <span m="1898080">of</span> <span m="1899250">tree.</span> <span m="1900030">And</span> <span m="1900950">what</span> <span m="1901120">you</span> <span m="1901280">want</span> <span m="1901460">to</span> <span m="1901580">prove,</span> <span m="1902220">first</span> <span m="1902560">of</span> <span m="1902600">all,</span> <span m="1902950">is</span> <span m="1903250">that</span> <span m="1903710">for</span> <span m="1904000">every</span> <span m="1904360">connected</span> <span m="1904810">graph,</span> <span m="1905510">we</span> <span m="1905620">can</span> <span m="1905770">construct</span> <span m="1906220">such</span> <span m="1906610">a</span> <span m="1906750">spanning</span> <span m="1907190">tree.</span> </p>
<p><span m="1907870">So</span> <span m="1908040">let's</span> <span m="1908230">first</span> <span m="1908440">define</span> <span m="1908830">this.</span> <span m="1909660">And</span> <span m="1909900">then</span> <span m="1911136">we</span> <span m="1911580">will</span> <span m="1911730">prove</span> <span m="1912120">this</span> <span m="1912350">theorem.</span> </p>
<p></p>
<p><span m="1916530">And</span> <span m="1916650">then</span> <span m="1916790">after</span> <span m="1917080">this,</span> <span m="1917430">we're</span> <span m="1917750">going to</span> <span m="1917870">talk</span> <span m="1918480">about</span> <span m="1919170">weighted</span> <span m="1919580">graphs.</span> <span m="1920660">That</span> <span m="1920870">will</span> <span m="1920980">lead</span> <span m="1921380">to</span> <span m="1922870">minimum</span> <span m="1923310">weight</span> <span m="1923640">spanning</span> <span m="1924030">trees.</span> <span m="1924800">And</span> <span m="1924930">then</span> <span m="1925020">it</span> <span m="1925170">gets</span> <span m="1925410">really</span> <span m="1925660">interesting</span> <span m="1926230">because</span> <span m="1926460">you</span> <span m="1926590">want</span> <span m="1926740">to</span> <span m="1926830">figure</span> <span m="1927170">out</span> <span m="1927460">how</span> <span m="1927610">we</span> <span m="1927740">can</span> <span m="1927960">actually</span> <span m="1928510">construct</span> <span m="1928980">those</span> <span m="1929780">for</span> <span m="1929990">any</span> <span m="1930210">graph.</span> </p>
<p></p>
<p><span m="1935830">So</span> <span m="1936060">let's</span> <span m="1936600">define a</span> <span m="1937040">spanning</span> <span m="1937410">tree.</span> <span m="1939930">A</span> <span m="1940360">spanning</span> <span m="1940860">tree--</span> </p>
<p></p>
<p><span m="1947140">And</span> <span m="1947650">we</span> <span m="1947810">appreciate</span> <span m="1948480">this</span> <span m="1949590">as</span> <span m="1950040">SD--</span> <span m="1951800">of</span> <span m="1952230">a</span> <span m="1954680">connected</span> <span m="1955470">graph</span> <span m="1960230">is</span> <span m="1960500">actually</span> <span m="1961740">a</span> <span m="1961870">subgraph</span> <span m="1969370">that</span> <span m="1969540">is</span> <span m="1969720">a</span> <span m="1969830">tree.</span> <span m="1971820">So</span> <span m="1971970">that's</span> <span m="1972180">the</span> <span m="1972250">first</span> <span m="1972530">condition.</span> <span m="1973900">And</span> <span m="1974150">it</span> <span m="1974260">also</span> <span m="1974590">covers</span> <span m="1974790">all</span> <span m="1974940">the</span> <span m="1975080">other</span> <span m="1975230">vertices.</span> </p>
<p><span m="1975940">So</span> <span m="1976650">it</span> <span m="1976830">has</span> <span m="1978160">the</span> <span m="1978280">same</span> <span m="1978750">vertices</span> <span m="1985860">as</span> <span m="1986340">the</span> <span m="1986440">graph.</span> </p>
<p></p>
<p><span m="1993520">And</span> <span m="1994040">over</span> <span m="1994310">here,</span> <span m="1994440">we</span> <span m="1994530">have</span> <span m="1994720">this</span> <span m="1994880">example.</span> </p>
<p><span m="1996970">So</span> <span m="1997160">now</span> <span m="1997360">what</span> <span m="1997510">you</span> <span m="1997630">want</span> <span m="1997770">to</span> <span m="1997860">show</span> <span m="1998250">is</span> <span m="1998690">the</span> <span m="1998780">following</span> <span m="1999110">theorem</span> <span m="1999960">that</span> <span m="2000210">any</span> <span m="2000550">connected</span> <span m="2000990">graph</span> <span m="2001470">actually</span> <span m="2001940">has</span> <span m="2002240">such</span> <span m="2002770">a</span> <span m="2002850">spanning</span> <span m="2003250">tree.</span> <span m="2004810">So</span> <span m="2006380">the</span> <span m="2006490">theorem</span> <span m="2007000">is</span> <span m="2010930">every</span> <span m="2011270">connected</span> <span m="2019320">graph</span> <span m="2023110">has</span> <span m="2023240">a</span> <span m="2024000">spanning</span> <span m="2024450">tree.</span> <span m="2025760">And</span> <span m="2027200">let's</span> <span m="2028620">think</span> <span m="2028890">about</span> <span m="2029140">a proof</span> <span m="2029460">together.</span> </p>
<p><span m="2032090">So</span> <span m="2032860">for</span> <span m="2033080">example,</span> <span m="2035170">we</span> <span m="2035710">may</span> <span m="2037010">start</span> <span m="2037560">with</span> <span m="2038500">the</span> <span m="2038630">idea</span> <span m="2039020">to</span> <span m="2039190">use</span> <span m="2039830">a</span> <span m="2039950">contradiction.</span> <span m="2041510">Suppose</span> <span m="2042190">there</span> <span m="2042360">exists</span> <span m="2043420">a</span> <span m="2043630">connected</span> <span m="2044030">graph</span> <span m="2044630">that</span> <span m="2044810">has</span> <span m="2045020">no</span> <span m="2045190">spanning</span> <span m="2045550">tree.</span> <span m="2046450">So</span> <span m="2046630">how</span> <span m="2046800">would</span> <span m="2046950">we</span> <span m="2047040">go</span> <span m="2047260">ahead?</span> </p>
<p></p>
<p><span m="2055850">So</span> <span m="2056050">how</span> <span m="2056239">can</span> <span m="2056400">we</span> <span m="2056530">prove</span> <span m="2057130">that</span> <span m="2060400">we</span> <span m="2060520">have a</span> <span m="2061000">contradiction</span> <span m="2061360">here?</span> <span m="2061644">So</span> <span m="2061929">the</span> <span m="2062010">proof</span> <span m="2062840">that</span> <span m="2062969">we're</span> <span m="2063090">going</span> <span m="2063300">to</span> <span m="2063480">propose</span> <span m="2063980">here</span> <span m="2064830">is</span> <span m="2065120">by</span> <span m="2065330">contradiction.</span> </p>
<p></p>
<p><span m="2073320">That</span> <span m="2073469">means</span> <span m="2073719">that</span> <span m="2073909">we're going</span> <span m="2074100">to</span> <span m="2074300">assume</span> <span m="2074540">the</span> <span m="2074699">opposite.</span> <span m="2075620">So</span> <span m="2075840">assume</span> <span m="2078120">that</span> <span m="2078880">we</span> <span m="2079040">have</span> <span m="2079429">a</span> <span m="2079580">connected</span> <span m="2080120">graph,</span> <span m="2082040">a</span> <span m="2082980">connected</span> <span m="2083489">graph</span> <span m="2084250">G.</span> </p>
<p></p>
<p></p>
<p><span m="2094570">That</span> <span m="2094780">has</span> <span m="2095150">no</span> <span m="2095550">spanning</span> <span m="2096030">tree.</span> <span m="2098600">So</span> <span m="2098790">how</span> <span m="2098950">can</span> <span m="2099120">we</span> <span m="2099630">use</span> <span m="2100010">this?</span> <span m="2101450">How</span> <span m="2101680">could</span> <span m="2101870">this</span> <span m="2102120">be</span> <span m="2102340">strange?</span> <span m="2103300">Or</span> <span m="2104580">what</span> <span m="2104710">can</span> <span m="2104830">we</span> <span m="2104940">do</span> <span m="2105130">here?</span> </p>
<p></p>
<p><span m="2113420">So</span> <span m="2113570">one</span> <span m="2113750">of</span> <span m="2113810">the</span> <span m="2113960">nice</span> <span m="2114360">things</span> <span m="2114560">about</span> <span m="2115890">trees</span> <span m="2116220">over</span> <span m="2116460">here</span> <span m="2116590">is--</span> <span m="2117390">so</span> <span m="2118180">that</span> <span m="2118320">maybe</span> <span m="2118620">gives</span> <span m="2118700">us</span> <span m="2118850">some</span> <span m="2119100">insight</span> <span m="2120220">is</span> <span m="2120470">that</span> <span m="2121860">if</span> <span m="2122210">we</span> <span m="2122660">have</span> <span m="2122970">a</span> <span m="2123050">tree--</span> <span m="2125330">well,</span> <span m="2126330">it</span> <span m="2126480">seems</span> <span m="2126940">like</span> <span m="2127910">such a</span> <span m="2128320">kind</span> <span m="2128500">of</span> <span m="2128560">spanning</span> <span m="2129000">tree</span> <span m="2129300">is like--</span> </p>
<p></p>
<p><span m="2132238">It's a</span> <span m="2132700">solution</span> <span m="2133460">to</span> <span m="2133560">a</span> <span m="2133850">subgraph</span> <span m="2134650">that</span> <span m="2134760">is connected</span> <span m="2136000">with</span> <span m="2136370">a</span> <span m="2136530">minimum</span> <span m="2136760">number</span> <span m="2137070">of</span> <span m="2137200">edges.</span> </p>
<p></p>
<p><span m="2140140">Like</span> <span m="2140530">if</span> <span m="2140740">I</span> <span m="2141370">have</span> <span m="2141720">any</span> <span m="2142010">other</span> <span m="2142920">type</span> <span m="2143360">of</span> <span m="2145200">subgraph--</span> <span m="2146340">for</span> <span m="2146500">example,</span> <span m="2146970">if</span> <span m="2147120">it has</span> <span m="2147350">cycles,</span> <span m="2147760">I can</span> <span m="2148020">always</span> <span m="2148300">remove</span> <span m="2148650">an</span> <span m="2148780">edge</span> <span m="2149000">of</span> <span m="2149120">course.</span> <span m="2149900">So</span> <span m="2151990">we</span> <span m="2152140">know</span> <span m="2153230">a</span> <span m="2153330">subgraph</span> <span m="2154660">of</span> <span m="2155160">G</span> <span m="2155890">that</span> <span m="2158270">touches</span> <span m="2158660">all</span> <span m="2158860">the</span> <span m="2159050">different</span> <span m="2159440">edges</span> <span m="2160260">and</span> <span m="2161850">has</span> <span m="2162010">a</span> <span m="2162070">cycle</span> <span m="2163060">can</span> <span m="2163260">get</span> <span m="2164270">shortened</span> <span m="2164980">by</span> <span m="2165590">eliminating</span> <span m="2166280">one</span> <span m="2166510">edge</span> <span m="2166820">of</span> <span m="2167010">a</span> <span m="2167070">cycle</span> <span m="2168010">that's</span> <span m="2168240">within</span> <span m="2168540">such</span> <span m="2168780">as</span> <span m="2168940">a</span> <span m="2169080">subgraph.</span> </p>
<p><span m="2169370">So</span> <span m="2169710">maybe</span> <span m="2170130">we</span> <span m="2170490">have</span> <span m="2170690">an</span> <span m="2170780">idea</span> <span m="2171140">here.</span> <span m="2171910">And</span> <span m="2172920">maybe</span> <span m="2173230">we can</span> <span m="2173870">use</span> <span m="2174140">this.</span> <span m="2174340">So</span> <span m="2174530">maybe</span> <span m="2174820">we</span> <span m="2174940">can</span> <span m="2175110">say,</span> <span m="2175840">let's</span> <span m="2176200">set</span> <span m="2176420">consider</span> <span m="2179300">a</span> <span m="2180800">graph</span> <span m="2181430">T.</span> <span m="2183110">So</span> <span m="2183370">let</span> <span m="2183630">T</span> <span m="2184520">be</span> <span m="2185950">a</span> <span m="2186090">connected</span> <span m="2186750">subgraph</span> <span m="2187650">of</span> <span m="2187840">G,</span> <span m="2189330">but</span> <span m="2189550">with</span> <span m="2189680">a</span> <span m="2189730">property</span> <span m="2190730">that</span> <span m="2190840">it has</span> <span m="2191130">a</span> <span m="2191220">minimum</span> <span m="2191580">number</span> <span m="2191840">of</span> <span m="2192300">edges.</span> </p>
<p><span m="2194570">So</span> <span m="2194670">let this</span> <span m="2194970">be a</span> <span m="2195140">connected</span> <span m="2195550">subgraph</span> <span m="2200370">of</span> <span m="2200630">G.</span> <span m="2201790">And</span> <span m="2203000">we</span> <span m="2203160">assume</span> <span m="2204690">that</span> <span m="2205040">it</span> <span m="2205190">has</span> <span m="2205430">the</span> <span m="2205510">same</span> <span m="2205840">vertices</span> <span m="2206440">of</span> <span m="2206660">G</span> <span m="2206880">of</span> <span m="2207000">course.</span> </p>
<p></p>
<p><span m="2215050">In</span> <span m="2215120">addition,</span> <span m="2216920">it</span> <span m="2216990">has</span> <span m="2218180">the</span> <span m="2218300">smallest</span> <span m="2218910">number</span> <span m="2219240">of</span> <span m="2219500">edges</span> <span m="2219880">possible.</span> </p>
<p></p>
<p><span m="2237350">So</span> <span m="2237580">what</span> <span m="2237700">do</span> <span m="2237800">we</span> <span m="2237940">know?</span> <span m="2238720">Well,</span> <span m="2238830">let's</span> <span m="2239240">look</span> <span m="2239520">at</span> <span m="2239600">our</span> <span m="2239790">assumption</span> <span m="2240310">here.</span> <span m="2241410">We</span> <span m="2241560">said</span> <span m="2241970">that</span> <span m="2242200">a</span> <span m="2242330">connected</span> <span m="2242880">graph,</span> <span m="2244170">G--</span> <span m="2244860">that's</span> <span m="2245070">what</span> <span m="2245190">we</span> <span m="2245290">consider</span> <span m="2245700">here--</span> <span m="2246380">for</span> <span m="2246600">which</span> <span m="2248140">there</span> <span m="2248290">is</span> <span m="2248580">no</span> <span m="2248750">spanning</span> <span m="2249170">tree.</span> </p>
<p><span m="2249390">So</span> <span m="2249570">what</span> <span m="2249670">do</span> <span m="2249750">we</span> <span m="2249860">know?</span> <span m="2250170">We</span> <span m="2250270">know</span> <span m="2250350">that</span> <span m="2250530">this</span> <span m="2250700">particular</span> <span m="2251840">T</span> <span m="2252090">over</span> <span m="2252460">here</span> <span m="2253000">can</span> <span m="2253305">definitely</span> <span m="2253610">not be a</span> <span m="2253800">spanning</span> <span m="2254210">tree,</span> <span m="2254420">because</span> <span m="2254660">that's</span> <span m="2254830">what</span> <span m="2254950">we</span> <span m="2255060">assume</span> <span m="2255510">here.</span> </p>
<p></p>
<p><span m="2259890">Can</span> <span m="2260080">we</span> <span m="2260130">use</span> <span m="2260480">this?</span> <span m="2262920">So</span> <span m="2262990">let's</span> <span m="2263200">think</span> <span m="2263410">about</span> <span m="2263730">this.</span> </p>
<p></p>
<p><span m="2269780">So</span> <span m="2270070">if</span> <span m="2270210">it</span> <span m="2270320">is</span> <span m="2270510">not</span> <span m="2272150">a</span> <span m="2272250">spanning</span> <span m="2272700">tree,</span> <span m="2273610">then</span> <span m="2274100">what</span> <span m="2274360">must it</span> <span m="2274690">have?</span> <span m="2275510">So</span> <span m="2275750">we</span> <span m="2276660">already</span> <span m="2277020">know</span> <span m="2277840">that</span> <span m="2278830">T</span> <span m="2279120">is</span> <span m="2279370">a</span> <span m="2279450">connected</span> <span m="2280020">subgraph</span> <span m="2280580">of</span> <span m="2280830">G</span> <span m="2281380">with</span> <span m="2281550">the</span> <span m="2281640">same</span> <span m="2281900">vertices</span> <span m="2282370">of</span> <span m="2283350">G.</span> <span m="2284610">So</span> <span m="2285140">the</span> <span m="2285260">only</span> <span m="2285550">difference</span> <span m="2286180">between</span> <span m="2287680">a</span> <span m="2288060">spanning</span> <span m="2288570">tree</span> <span m="2289180">and</span> <span m="2289430">T</span> <span m="2289870">is</span> <span m="2290070">that</span> <span m="2290180">the</span> <span m="2290260">spanning</span> <span m="2290670">tree</span> <span m="2290940">is a</span> <span m="2291130">tree,</span> <span m="2292250">and</span> <span m="2294000">T</span> <span m="2294290">is not</span> <span m="2294440">a</span> <span m="2294540">tree.</span> <span m="2295550">So</span> <span m="2295710">therefor,</span> <span m="2296080">T</span> <span m="2296410">must have</span> <span m="2296570">a</span> <span m="2296620">cycle.</span> </p>
<p><span m="2297870">So</span> <span m="2299800">we</span> <span m="2300130">use</span> <span m="2300460">our</span> <span m="2300680">assumption</span> <span m="2301160">over</span> <span m="2301440">here.</span> <span m="2302790">We</span> <span m="2302890">know</span> <span m="2303120">that</span> <span m="2303260">T</span> <span m="2303550">is</span> <span m="2303750">not</span> <span m="2305480">a</span> <span m="2305610">spanning</span> <span m="2306030">tree.</span> <span m="2307730">And</span> <span m="2308030">now we can</span> <span m="2308150">conclude</span> <span m="2310040">that</span> <span m="2310160">it</span> <span m="2310290">must</span> <span m="2310540">have</span> <span m="2310700">a</span> <span m="2310790">cycle.</span> </p>
<p></p>
<p><span m="2316960">So</span> <span m="2317190">now</span> <span m="2317440">we</span> <span m="2317580">can</span> <span m="2318970">start</span> <span m="2320300">thinking</span> <span m="2320730">about</span> <span m="2321210">this</span> <span m="2321910">cycle.</span> <span m="2322530">And</span> <span m="2322810">let's</span> <span m="2323280">feel--</span> </p>
<p></p>
<p><span m="2326540">What</span> <span m="2326690">can</span> <span m="2326790">we</span> <span m="2326900">do</span> <span m="2327050">here?</span> <span m="2327830">So</span> <span m="2329710">we</span> <span m="2330050">constructed</span> <span m="2330720">T</span> <span m="2331100">as</span> <span m="2331620">the</span> <span m="2331730">subgraph</span> <span m="2332380">that has</span> <span m="2332510">the</span> <span m="2333570">minimum</span> <span m="2333970">number</span> <span m="2334290">of</span> <span m="2334750">edges</span> <span m="2334890">possible.</span> <span m="2335980">So</span> <span m="2336130">now</span> <span m="2336280">if</span> <span m="2336390">you</span> <span m="2336470">have</span> <span m="2336560">a</span> <span m="2336620">cycle,</span> <span m="2337080">the</span> <span m="2337140">whole</span> <span m="2337410">idea</span> <span m="2337480">is</span> <span m="2337760">that</span> <span m="2337910">we're</span> <span m="2337980">going</span> <span m="2338180">to</span> <span m="2338930">delete</span> <span m="2339350">one</span> <span m="2339620">of</span> <span m="2339760">the</span> <span m="2339880">edges</span> <span m="2340210">of</span> <span m="2340360">the</span> <span m="2340430">cycle.</span> </p>
<p><span m="2341750">And</span> <span m="2341920">if</span> <span m="2342010">you</span> <span m="2342100">do</span> <span m="2342290">that,</span> <span m="2343040">we will</span> <span m="2343270">be</span> <span m="2343420">able</span> <span m="2343860">to</span> <span m="2344230">construct</span> <span m="2344790">a</span> <span m="2344940">smaller</span> <span m="2346710">subgraph</span> <span m="2347020">T,</span> <span m="2347820">essentially,</span> <span m="2349310">with</span> <span m="2350080">a</span> <span m="2350130">smaller</span> <span m="2350540">number</span> <span m="2350810">of</span> <span m="2350960">edges.</span> <span m="2351420">And that</span> <span m="2351540">contradicts</span> <span m="2353160">this</span> <span m="2353410">particular</span> <span m="2353830">statement</span> <span m="2354330">over</span> <span m="2354600">here.</span> <span m="2355370">So</span> <span m="2355520">that's</span> <span m="2355730">sort</span> <span m="2355950">of</span> <span m="2356040">how</span> <span m="2356260">we</span> <span m="2356350">go</span> <span m="2356590">ahead.</span> </p>
<p><span m="2357900">So</span> <span m="2358330">if</span> <span m="2358490">it has</span> <span m="2358800">a</span> <span m="2358850">cycle,</span> <span m="2360110">let's</span> <span m="2360640">write</span> <span m="2360890">it</span> <span m="2361030">out.</span> <span m="2361370">For</span> <span m="2361520">example,</span> <span m="2362400">here</span> <span m="2362660">we</span> <span m="2362760">have</span> <span m="2362980">a</span> <span m="2363060">cycle</span> <span m="2365700">like</span> <span m="2365940">this.</span> </p>
<p></p>
<p><span m="2368870">Well,</span> <span m="2369580">let's</span> <span m="2370720">prove</span> <span m="2371580">that</span> <span m="2373040">if</span> <span m="2373300">we</span> <span m="2373620">remove</span> <span m="2374690">one</span> <span m="2375410">of</span> <span m="2375560">the</span> <span m="2375650">cycle,</span> <span m="2376820">that we</span> <span m="2377010">still</span> <span m="2377530">have</span> <span m="2378040">a</span> <span m="2379420">connected</span> <span m="2379850">subgraph.</span> <span m="2381280">So</span> <span m="2381470">the</span> <span m="2381550">whole</span> <span m="2381830">graph</span> <span m="2382233">T</span> <span m="2383040">is</span> <span m="2383200">much</span> <span m="2383470">bigger,</span> <span m="2383780">right?</span> <span m="2384180">It</span> <span m="2384455">has</span> <span m="2384730">lots</span> <span m="2385140">of</span> <span m="2386090">connections</span> <span m="2387170">and</span> <span m="2387350">so</span> <span m="2387560">on.</span> </p>
<p><span m="2389480">Suppose</span> <span m="2389930">we will</span> <span m="2390270">remove</span> <span m="2390740">this</span> <span m="2390930">particular</span> <span m="2391610">edge</span> <span m="2392060">over</span> <span m="2392350">here.</span> <span m="2393470">So</span> <span m="2393700">we</span> <span m="2394100">remove</span> <span m="2394540">this</span> <span m="2394760">one.</span> <span m="2397650">Do</span> <span m="2397790">we</span> <span m="2397950">still</span> <span m="2398730">have</span> <span m="2400040">a</span> <span m="2400950">connected</span> <span m="2401530">subgraph</span> <span m="2402410">of</span> <span m="2402610">G</span> <span m="2403060">that</span> <span m="2403450">covers</span> <span m="2403740">all</span> <span m="2404270">the</span> <span m="2404370">vertices</span> <span m="2405210">of</span> <span m="2405450">G?</span> </p>
<p><span m="2406060">Well, of</span> <span m="2406300">course it</span> <span m="2406690">cover</span> <span m="2407010">all</span> <span m="2407190">the</span> <span m="2407260">vertices</span> <span m="2407680">of</span> <span m="2407820">G</span> <span m="2408010">because</span> <span m="2408260">we</span> <span m="2408380">only</span> <span m="2408640">removed</span> <span m="2408920">an</span> <span m="2409020">edge</span> <span m="2409330">over</span> <span m="2409600">here.</span> <span m="2409980">So</span> <span m="2410140">that's</span> <span m="2410720">fine.</span> <span m="2411630">All</span> <span m="2411830">these</span> <span m="2412070">parts</span> <span m="2412380">are</span> <span m="2412490">still</span> <span m="2412740">connected</span> <span m="2413180">to</span> <span m="2413280">the</span> <span m="2413410">rest.</span> </p>
<p><span m="2416910">If</span> <span m="2417820">you</span> <span m="2417890">can</span> <span m="2418060">show</span> <span m="2418360">that</span> <span m="2418510">it is</span> <span m="2418570">connected,</span> <span m="2419906">then</span> <span m="2420260">well</span> <span m="2420570">we</span> <span m="2420790">just--</span> <span m="2423080">well,</span> <span m="2423260">we</span> <span m="2423400">removed</span> <span m="2423780">one</span> <span m="2424010">edge.</span> <span m="2424380">So</span> <span m="2424530">we</span> <span m="2424620">actually</span> <span m="2425040">created</span> <span m="2425380">this</span> <span m="2425520">smaller</span> <span m="2426660">graph.</span> <span m="2427210">And</span> <span m="2427350">then</span> <span m="2427480">we</span> <span m="2427550">get</span> <span m="2427720">contradiction.</span> <span m="2428510">And then</span> <span m="2428710">we're done</span> <span m="2428920">with</span> <span m="2429280">our</span> <span m="2429530">proof.</span> </p>
<p><span m="2430800">So</span> <span m="2430960">let's</span> <span m="2431320">see</span> <span m="2432060">whether</span> <span m="2432280">we</span> <span m="2432380">can</span> <span m="2432540">prove</span> <span m="2433480">that</span> <span m="2434320">if</span> <span m="2434600">we</span> <span m="2434790">remove</span> <span m="2435330">this</span> <span m="2435500">particular</span> <span m="2435910">edge</span> <span m="2436110">over</span> <span m="2436330">here,</span> <span m="2436750">that</span> <span m="2436870">we</span> <span m="2437000">still</span> <span m="2437350">have a</span> <span m="2437550">connected</span> <span m="2437970">subgraph.</span> <span m="2439110">So</span> <span m="2439270">how</span> <span m="2439420">can</span> <span m="2439550">we</span> <span m="2439650">do</span> <span m="2439800">this?</span> </p>
<p><span m="2442760">Well,</span> <span m="2443080">let's</span> <span m="2443510">take--</span> <span m="2445800">so</span> <span m="2445960">the</span> <span m="2446050">first</span> <span m="2446490">case</span> <span m="2447530">would</span> <span m="2447750">be</span> <span m="2449010">if</span> <span m="2450190">a</span> <span m="2450580">vertex</span> <span m="2451382">x</span> <span m="2452146">is</span> <span m="2452530">connected</span> <span m="2455020">to</span> <span m="2455810">y.</span> <span m="2456020">So</span> <span m="2456360">I</span> <span m="2456460">want</span> <span m="2456610">to</span> <span m="2456720">take</span> <span m="2456990">any</span> <span m="2457360">pair</span> <span m="2457810">x,</span> <span m="2457910">y</span> <span m="2458420">of</span> <span m="2458520">vertices.</span> <span m="2460000">And</span> <span m="2460450">suppose</span> <span m="2461050">that</span> <span m="2461390">the</span> <span m="2462020">path</span> <span m="2462335">that</span> <span m="2462650">connects</span> <span m="2463160">x</span> <span m="2463510">and</span> <span m="2463770">y</span> <span m="2464210">does</span> <span m="2464430">not</span> <span m="2464740">contain</span> <span m="2465280">this</span> <span m="2465480">particular</span> <span m="2466620">edge</span> <span m="2467040">e</span> <span m="2467430">that I</span> <span m="2467730">remove</span> <span m="2468170">here.</span> <span m="2469030">So</span> <span m="2469190">suppose</span> <span m="2469680">this</span> <span m="2469880">does</span> <span m="2470060">not</span> <span m="2473510">contain</span> <span m="2476080">e.</span> </p>
<p><span m="2478000">So</span> <span m="2478170">I</span> <span m="2478250">look</span> <span m="2478500">at</span> <span m="2479210">the</span> <span m="2479280">graph</span> <span m="2479620">T.</span> <span m="2480432">I</span> <span m="2480840">Know</span> <span m="2481190">T</span> <span m="2481490">is</span> <span m="2481640">connected.</span> <span m="2482770">So</span> <span m="2482980">I</span> <span m="2483070">take</span> <span m="2483380">a</span> <span m="2483440">pair</span> <span m="2483720">of</span> <span m="2483820">vertices</span> <span m="2484440">x</span> <span m="2484690">and</span> <span m="2484850">y.</span> <span m="2485110">I</span> <span m="2485230">know</span> <span m="2485450">there</span> <span m="2485650">is</span> <span m="2485790">a</span> <span m="2485870">path</span> <span m="2486620">between</span> <span m="2487120">x</span> <span m="2487450">and</span> <span m="2487730">y</span> <span m="2488490">in</span> <span m="2488850">T.</span> <span m="2491020">If</span> <span m="2491670">this</span> <span m="2492010">path</span> <span m="2492500">does not</span> <span m="2492750">contain</span> <span m="2493780">the</span> <span m="2493900">edge</span> <span m="2494335">e,</span> <span m="2495360">then</span> <span m="2495540">I</span> <span m="2495620">know</span> <span m="2496040">that</span> <span m="2496200">the</span> <span m="2496290">same</span> <span m="2496660">path</span> <span m="2496980">still</span> <span m="2497300">exists</span> <span m="2498360">in</span> <span m="2498520">the</span> <span m="2498620">same</span> <span m="2498960">graph</span> <span m="2499410">where</span> <span m="2499600">I've</span> <span m="2499830">removed this</span> <span m="2500290">particular</span> <span m="2500800">edge.</span> </p>
<p><span m="2501620">So</span> <span m="2501780">that's</span> <span m="2502060">great,</span> <span m="2502370">right?</span> <span m="2503960">I have</span> <span m="2504240">shown</span> <span m="2504760">that</span> <span m="2504990">x</span> <span m="2505250">and</span> <span m="2505410">y</span> <span m="2505660">are</span> <span m="2505790">still</span> <span m="2506110">connected</span> <span m="2507110">in</span> <span m="2507310">the</span> <span m="2507740">new</span> <span m="2508080">graph</span> <span m="2508375">where</span> <span m="2508670">I've</span> <span m="2509070">removed</span> <span m="2509550">this</span> <span m="2509790">e.</span> </p>
<p><span m="2511250">So</span> <span m="2511410">now</span> <span m="2511590">let's</span> <span m="2511870">look</span> <span m="2512040">at</span> <span m="2512120">another</span> <span m="2512450">case.</span> <span m="2512870">For</span> <span m="2512960">example,</span> <span m="2514600">x</span> <span m="2514860">is</span> <span m="2515050">over</span> <span m="2515360">here.</span> <span m="2516440">And</span> <span m="2516630">it</span> <span m="2516780">is</span> <span m="2516940">connected</span> <span m="2519650">all</span> <span m="2519880">the</span> <span m="2519980">way</span> <span m="2520170">through</span> <span m="2520490">here</span> <span m="2521500">over</span> <span m="2521640">this</span> <span m="2521820">particular</span> <span m="2522330">edge</span> <span m="2524060">all</span> <span m="2524250">the</span> <span m="2524360">way</span> <span m="2524560">to,</span> <span m="2525260">say,</span> <span m="2525570">y</span> <span m="2525810">over</span> <span m="2526110">here.</span> <span m="2526770">So</span> <span m="2526900">this</span> <span m="2527090">is</span> <span m="2527290">case</span> <span m="2527620">number</span> <span m="2527880">two.</span> </p>
<p></p>
<p><span m="2535720">Well,</span> <span m="2537340">are</span> <span m="2537610">x</span> <span m="2537875">ad y</span> <span m="2538140">still</span> <span m="2538410">connected?</span> <span m="2539660">Yes</span> <span m="2539910">they</span> <span m="2540110">are,</span> <span m="2540300">right?</span> <span m="2540650">Because</span> <span m="2541680">I</span> <span m="2542030">have</span> <span m="2542130">removed</span> <span m="2542460">e</span> <span m="2543816">from</span> <span m="2544720">a</span> <span m="2544830">cycle.</span> <span m="2545610">So</span> <span m="2546240">I</span> <span m="2546330">can</span> <span m="2546540">still</span> <span m="2547070">go</span> <span m="2548310">into</span> <span m="2548540">the</span> <span m="2548670">other</span> <span m="2548790">direction</span> <span m="2549250">of</span> <span m="2549400">the</span> <span m="2549480">cycle</span> <span m="2550400">and</span> <span m="2550620">connect</span> <span m="2551720">back</span> <span m="2553220">towards</span> <span m="2553630">y.</span> </p>
<p><span m="2554540">So</span> <span m="2555500">what</span> <span m="2555680">we</span> <span m="2555780">see</span> <span m="2556080">here</span> <span m="2556310">is</span> <span m="2556460">that</span> <span m="2556600">x</span> <span m="2556980">can</span> <span m="2557270">go</span> <span m="2557510">all</span> <span m="2557740">the</span> <span m="2557840">way</span> <span m="2558200">to</span> <span m="2558370">here.</span> <span m="2558710">But</span> <span m="2558850">it</span> <span m="2558970">can</span> <span m="2559140">still</span> <span m="2559470">go</span> <span m="2560110">over</span> <span m="2560330">the</span> <span m="2560440">cycle</span> <span m="2560880">back</span> <span m="2561110">to</span> <span m="2561230">here,</span> <span m="2562010">and</span> <span m="2562150">then</span> <span m="2562460">to</span> <span m="2562745">y.</span> <span m="2565090">So</span> <span m="2565700">we</span> <span m="2565940">are</span> <span m="2566050">still</span> <span m="2566330">connected.</span> </p>
<p><span m="2568410">So</span> <span m="2568520">this</span> <span m="2568710">shows</span> <span m="2569260">that</span> <span m="2569700">in</span> <span m="2570040">both</span> <span m="2570340">of</span> <span m="2570450">the</span> <span m="2570570">cases,</span> <span m="2571120">for</span> <span m="2571330">any</span> <span m="2571610">pair</span> <span m="2571840">of</span> <span m="2571940">vertices,</span> <span m="2572960">even</span> <span m="2573440">after</span> <span m="2573850">removal</span> <span m="2574470">of</span> <span m="2574810">e,</span> <span m="2575260">the</span> <span m="2575360">graph</span> <span m="2575720">T</span> <span m="2576670">minus</span> <span m="2577350">this</span> <span m="2577520">edge</span> <span m="2577700">e</span> <span m="2578200">is</span> <span m="2578350">still</span> <span m="2578690">connected.</span> </p>
<p><span m="2581650">So</span> <span m="2581800">let's</span> <span m="2582010">write</span> <span m="2582190">this</span> <span m="2582420">out.</span> <span m="2582990">So</span> <span m="2583980">all</span> <span m="2584550">vertices</span> <span m="2589890">in</span> <span m="2590120">G</span> <span m="2595110">are</span> <span m="2595320">still</span> <span m="2595720">connected</span> <span m="2600540">after</span> <span m="2601140">removing</span> <span m="2607700">e</span> <span m="2608310">from</span> <span m="2609130">T.</span> </p>
<p><span m="2611150">So</span> <span m="2611440">now</span> <span m="2611620">we</span> <span m="2611740">are</span> <span m="2611790">done.</span> <span m="2612730">So</span> <span m="2612870">what</span> <span m="2613020">was</span> <span m="2613220">that</span> <span m="2613370">again?</span> <span m="2615320">We</span> <span m="2615450">now</span> <span m="2615680">constructed</span> <span m="2616300">a</span> <span m="2616380">smaller</span> <span m="2616910">graph--</span> <span m="2617290">smaller</span> <span m="2617740">in</span> <span m="2617840">the</span> <span m="2617920">sense</span> <span m="2618250">that</span> <span m="2618360">it</span> <span m="2618500">has</span> <span m="2619840">one</span> <span m="2620290">less</span> <span m="2620680">edge--</span> <span m="2621980">that</span> <span m="2622140">is</span> <span m="2622250">still</span> <span m="2622540">connected,</span> <span m="2624300">and</span> <span m="2624450">with</span> <span m="2624580">the</span> <span m="2624660">same</span> <span m="2624880">vertices</span> <span m="2625290">as</span> <span m="2625590">G.</span> <span m="2626890">But</span> <span m="2627000">I</span> <span m="2627090">assumed</span> <span m="2627700">that</span> <span m="2627970">T</span> <span m="2628090">was</span> <span m="2628290">already</span> <span m="2628560">the</span> <span m="2628680">smallest</span> <span m="2629470">such</span> <span m="2630250">graph,</span> <span m="2631440">such</span> <span m="2631600">subgraph.</span> <span m="2632450">So</span> <span m="2632610">we</span> <span m="2632690">have a</span> <span m="2632830">contradiction.</span> </p>
<p><span m="2634370">So</span> <span m="2634860">what</span> <span m="2634970">does</span> <span m="2635130">this</span> <span m="2635350">mean?</span> <span m="2636190">This</span> <span m="2636260">means</span> <span m="2636530">that</span> <span m="2636650">our</span> <span m="2636970">original</span> <span m="2637470">assumption</span> <span m="2638090">that</span> <span m="2638230">we</span> <span m="2638300">have</span> <span m="2638530">over</span> <span m="2638870">here</span> <span m="2639780">cannot</span> <span m="2640080">be</span> <span m="2640200">true</span> <span m="2641690">because</span> <span m="2641970">we</span> <span m="2642070">started</span> <span m="2642460">off</span> <span m="2642770">with</span> <span m="2642990">this</span> <span m="2643170">assumption.</span> <span m="2644530">Because</span> <span m="2644980">of</span> <span m="2645070">this,</span> <span m="2645630">we</span> <span m="2645840">could</span> <span m="2646040">start</span> <span m="2646420">to</span> <span m="2646590">construct</span> <span m="2647080">this.</span> <span m="2647460">And</span> <span m="2647620">then</span> <span m="2647760">you</span> <span m="2647870">could</span> <span m="2648220">derive</span> <span m="2648650">this</span> <span m="2648850">whole</span> <span m="2650700">argumentation.</span> <span m="2652070">And in the end,</span> <span m="2652450">we</span> <span m="2652560">get</span> <span m="2652740">a</span> <span m="2652790">contradiction.</span> </p>
<p><span m="2653330">So</span> <span m="2653440">our</span> <span m="2653840">original</span> <span m="2654290">assumption</span> <span m="2654760">is</span> <span m="2654890">wrong.</span> <span m="2655750">So</span> <span m="2656020">actually,</span> <span m="2659150">this</span> <span m="2659350">is</span> <span m="2659430">not</span> <span m="2659610">possible.</span> <span m="2660040">There</span> <span m="2660150">does</span> <span m="2660360">not</span> <span m="2660510">exist</span> <span m="2660830">a</span> <span m="2660940">connected</span> <span m="2661380">graph</span> <span m="2662020">G</span> <span m="2662310">that</span> <span m="2662450">has</span> <span m="2662760">no</span> <span m="2663050">ST.</span> <span m="2663760">So</span> <span m="2663930">the</span> <span m="2664030">theorem</span> <span m="2664460">is</span> <span m="2664620">true.</span> </p>
<p><span m="2667120">So</span> <span m="2667330">this</span> <span m="2667570">is</span> <span m="2668750">an</span> <span m="2669020">important</span> <span m="2669720">theorem.</span> <span m="2670590">So</span> <span m="2670850">now</span> <span m="2671090">we</span> <span m="2671240">can</span> <span m="2672000">start</span> <span m="2672440">talking</span> <span m="2672790">about</span> <span m="2673110">weighted</span> <span m="2673850">minimum</span> <span m="2674240">spanning</span> <span m="2674640">trees.</span> </p>
<p></p>
<p><span m="2680620">So</span> <span m="2680860">let's</span> <span m="2681580">use</span> <span m="2681870">this</span> <span m="2682310">picture</span> <span m="2682650">over</span> <span m="2682950">here</span> <span m="2685710">and</span> <span m="2686940">assign</span> <span m="2687340">some</span> <span m="2687590">weights</span> <span m="2688020">to</span> <span m="2688230">this</span> <span m="2688770">graph.</span> </p>
<p></p>
<p><span m="2692000">So</span> <span m="2692330">for</span> <span m="2692500">example,</span> <span m="2694840">we</span> <span m="2694960">have</span> <span m="2695840">1,</span> <span m="2697580">2,</span> <span m="2701050">2,</span> <span m="2703210">3,</span> <span m="2704614">3,</span> <span m="2706018">3,</span> <span m="2707290">a</span> <span m="2707670">1</span> <span m="2707890">over</span> <span m="2708170">here,</span> <span m="2708800">another</span> <span m="2709130">1,</span> <span m="2710050">1,</span> <span m="2711510">4,</span> <span m="2712730">and</span> <span m="2713030">7.</span> <span m="2715130">And</span> <span m="2715420">let's</span> <span m="2718090">construct</span> <span m="2719570">a</span> <span m="2719640">spanning</span> <span m="2720170">tree.</span> </p>
<p><span m="2720950">So</span> <span m="2721090">let</span> <span m="2721220">me</span> <span m="2721310">give</span> <span m="2721480">an</span> <span m="2721570">example</span> <span m="2722140">over</span> <span m="2722420">here.</span> <span m="2723110">Actually,</span> <span m="2723380">this</span> <span m="2723580">is</span> <span m="2723690">an</span> <span m="2723800">example</span> <span m="2724650">that</span> <span m="2724750">we</span> <span m="2724890">have</span> <span m="2725100">right her</span> <span m="2725980">in</span> <span m="2726420">the</span> <span m="2726880">thick</span> <span m="2727300">lines in</span> <span m="2727390">green.</span> </p>
<p><span m="2729580">So</span> <span m="2729810">what</span> <span m="2730010">is</span> <span m="2730230">the</span> <span m="2730710">weight</span> <span m="2731190">of the</span> <span m="2731310">spanning</span> <span m="2732100">tree</span> <span m="2732430">that</span> <span m="2732570">we</span> <span m="2732660">have</span> <span m="2732930">here?</span> <span m="2733580">So</span> <span m="2733710">we</span> <span m="2733810">simply</span> <span m="2734360">add</span> <span m="2734640">the</span> <span m="2734740">weights</span> <span m="2735170">of</span> <span m="2735470">all the</span> <span m="2735610">edges.</span> <span m="2735920">So</span> <span m="2736120">we</span> <span m="2736230">have</span> <span m="2736370">7,</span> <span m="2737430">3.</span> <span m="2738020">That</span> <span m="2738150">makes</span> <span m="2738480">10.</span> <span m="2738960">11,</span> <span m="2739920">14,</span> <span m="2741230">another</span> <span m="2742380">2,</span> <span m="2742822">16,</span> <span m="2743264">17,</span> <span m="2743706">18,</span> <span m="2744150">19.</span> <span m="2744800">So</span> <span m="2745060">the</span> <span m="2745210">weight</span> <span m="2746440">of</span> <span m="2746700">this</span> <span m="2746930">particular</span> <span m="2749090">spanning</span> <span m="2749620">tree</span> <span m="2751930">is</span> <span m="2752170">19.</span> </p>
<p><span m="2753650">So</span> <span m="2753690">now</span> <span m="2753870">the</span> <span m="2753990">problem</span> <span m="2754480">is--</span> <span m="2755390">and</span> <span m="2755510">that's</span> <span m="2755700">the</span> <span m="2755780">rest</span> <span m="2756020">of</span> <span m="2756160">the</span> <span m="2756230">lecture--</span> <span m="2757420">how</span> <span m="2757690">can</span> <span m="2757860">you</span> <span m="2757970">construct</span> <span m="2758480">a</span> <span m="2758560">minimum</span> <span m="2759110">weight</span> <span m="2759380">spanning</span> <span m="2759790">tree,</span> <span m="2760230">one</span> <span m="2761040">that</span> <span m="2761220">has</span> <span m="2762110">the</span> <span m="2762330">minimum</span> <span m="2763060">weight</span> <span m="2765260">possible?</span> </p>
<p><span m="2765870">So</span> <span m="2767140">we</span> <span m="2767310">see</span> <span m="2767480">another</span> <span m="2767740">example.</span> <span m="2769380">So</span> <span m="2769570">could</span> <span m="2769810">I</span> <span m="2769880">swap</span> <span m="2770230">some</span> <span m="2770440">edges</span> <span m="2770700">around</span> <span m="2771280">or</span> <span m="2771360">something</span> <span m="2771660">like</span> <span m="2771880">that</span> <span m="2772120">such</span> <span m="2772200">that</span> <span m="2772340">I</span> <span m="2772480">can</span> <span m="2772670">get</span> <span m="2773100">a</span> <span m="2773380">lesser</span> <span m="2773650">weight?</span> </p>
<p><span m="2775560">So</span> <span m="2776480">for</span> <span m="2776670">example,</span> <span m="2777300">over</span> <span m="2777550">here,</span> <span m="2777860">I</span> <span m="2777940">have--</span> <span m="2780580">for</span> <span m="2780600">example,</span> <span m="2780940">this</span> <span m="2781150">node</span> <span m="2781360">over</span> <span m="2781630">here</span> <span m="2782630">is</span> <span m="2782830">connected</span> <span m="2783480">by</span> <span m="2784310">an</span> <span m="2784530">edge</span> <span m="2784860">that</span> <span m="2785280">has</span> <span m="2785620">weight</span> <span m="2785920">3.</span> <span m="2787290">But</span> <span m="2787420">I</span> <span m="2787750">could</span> <span m="2787910">also</span> <span m="2788210">have</span> <span m="2789020">connected</span> <span m="2789305">it</span> <span m="2789590">to</span> <span m="2789830">this</span> <span m="2790030">particular</span> <span m="2790970">one</span> <span m="2791270">over</span> <span m="2791520">here,</span> <span m="2791900">which</span> <span m="2792000">is</span> <span m="2792160">only</span> <span m="2792450">weight</span> <span m="2792730">1.</span> </p>
<p><span m="2793530">If</span> <span m="2793700">I</span> <span m="2793750">do</span> <span m="2793910">this,</span> <span m="2795010">I</span> <span m="2795190">actually</span> <span m="2795760">improve</span> <span m="2796750">the</span> <span m="2796820">spanning</span> <span m="2797230">tree</span> <span m="2797460">construction.</span> <span m="2799220">So</span> <span m="2799590">I</span> <span m="2799660">will</span> <span m="2799910">get</span> <span m="2800050">something</span> <span m="2800590">that</span> <span m="2800770">looks</span> <span m="2801120">like--</span> </p>
<p></p>
<p><span m="2807480">Let's</span> <span m="2807950">see--</span> <span m="2809890">that</span> <span m="2810050">looks</span> <span m="2810390">like</span> <span m="2810700">this.</span> </p>
<p></p>
<p><span m="2814980">And</span> <span m="2816580">now</span> <span m="2817090">we</span> <span m="2817210">replaced</span> <span m="2817730">3</span> <span m="2818040">by</span> <span m="2818230">1.</span> <span m="2818590">So</span> <span m="2818810">it</span> <span m="2818950">gets</span> <span m="2819780">17.</span> </p>
<p><span m="2822260">So</span> <span m="2822450">can</span> <span m="2822600">we</span> <span m="2822810">do</span> <span m="2823030">anything</span> <span m="2823410">less</span> <span m="2823730">than</span> <span m="2823890">this?</span> <span m="2826060">That's</span> <span m="2826370">maybe--</span> <span m="2827175">it's</span> <span m="2827610">still</span> <span m="2827840">a</span> <span m="2827900">tree,</span> <span m="2828220">right?</span> <span m="2828580">Because</span> <span m="2828890">there's</span> <span m="2829080">no</span> <span m="2829210">cycle.</span> <span m="2829850">Here</span> <span m="2829990">we</span> <span m="2830090">have</span> <span m="2830360">this,</span> <span m="2830720">this,</span> <span m="2831000">this,</span> <span m="2831280">this part</span> <span m="2832620">over</span> <span m="2832770">here.</span> </p>
<p><span m="2834300">If</span> <span m="2834470">you</span> <span m="2834630">want</span> <span m="2834820">to have</span> <span m="2835130">connected</span> <span m="2835520">subgraph,</span> <span m="2836230">I</span> <span m="2836420">always</span> <span m="2836850">need</span> <span m="2837060">the</span> <span m="2837170">7</span> <span m="2837445">in</span> <span m="2837720">here.</span> </p>
<p></p>
<p><span m="2841490">Well,</span> <span m="2842140">it</span> <span m="2842300">seems</span> <span m="2842650">like</span> <span m="2842940">you cannot</span> <span m="2843140">really</span> <span m="2843430">replace</span> <span m="2844030">anything</span> <span m="2845170">by</span> <span m="2845560">something</span> <span m="2846080">cheaper.</span> <span m="2847020">Like</span> <span m="2847410">any</span> <span m="2847710">edge</span> <span m="2848120">in</span> <span m="2848300">here</span> <span m="2849910">seems</span> <span m="2850300">to</span> <span m="2850410">be</span> <span m="2850610">pretty</span> <span m="2853180">necessary.</span> </p>
<p><span m="2854490">Of</span> <span m="2854620">course,</span> <span m="2854860">what</span> <span m="2854980">I</span> <span m="2855030">could</span> <span m="2855230">do</span> <span m="2855410">is</span> <span m="2855550">I</span> <span m="2855600">could</span> <span m="2855880">create</span> <span m="2856300">other</span> <span m="2856520">minimum</span> <span m="2856840">spanning</span> <span m="2857280">trees.</span> <span m="2858000">So</span> <span m="2858130">actually it</span> <span m="2858490">turns</span> <span m="2858820">out</span> <span m="2858980">that</span> <span m="2859040">17</span> <span m="2859460">is</span> <span m="2859820">the</span> <span m="2859920">minimum</span> <span m="2860080">weight.</span> <span m="2861060">For</span> <span m="2861200">example,</span> <span m="2861650">I</span> <span m="2861750">could,</span> <span m="2862490">instead</span> <span m="2862760">of</span> <span m="2862870">this</span> <span m="2863180">edge,</span> <span m="2863640">I</span> <span m="2863720">could</span> <span m="2863910">use</span> <span m="2864230">this</span> <span m="2864575">edge.</span> <span m="2865270">That's</span> <span m="2865450">another</span> <span m="2865730">solution.</span> <span m="2866930">Or</span> <span m="2867410">instead</span> <span m="2867910">of</span> <span m="2868120">this</span> <span m="2868420">edge,</span> <span m="2868890">I</span> <span m="2868960">could</span> <span m="2869140">use</span> <span m="2869350">this</span> <span m="2869620">edge.</span> </p>
<p></p>
<p><span m="2877960">So</span> <span m="2878200">how</span> <span m="2878330">do</span> <span m="2878440">we</span> <span m="2878560">construct</span> <span m="2878930">such</span> <span m="2879130">a minimum</span> <span m="2879350">weight</span> <span m="2880370">spanning</span> <span m="2880840">tree?</span> <span m="2882080">Can</span> <span m="2882270">we</span> <span m="2883200">somehow</span> <span m="2883560">find</span> <span m="2883790">an</span> <span m="2883890">algorithm</span> <span m="2884530">that</span> <span m="2884790">creates</span> <span m="2885230">this?</span> <span m="2886330">And</span> <span m="2886630">that's</span> <span m="2887010">the</span> <span m="2887080">big</span> <span m="2887300">problem</span> <span m="2887680">of</span> <span m="2887780">today.</span> </p>
<p><span m="2889220">So</span> <span m="2889850">let</span> <span m="2890000">me</span> <span m="2890470">write</span> <span m="2890790">out</span> <span m="2891050">one</span> <span m="2891240">spanning</span> <span m="2891670">tree</span> <span m="2891860">that</span> <span m="2891970">I</span> <span m="2892100">will</span> <span m="2892290">use</span> <span m="2893070">for</span> <span m="2893340">the</span> <span m="2893470">rest</span> <span m="2894840">of</span> <span m="2895480">this</span> <span m="2895720">class.</span> <span m="2896890">And</span> <span m="2897040">maybe</span> <span m="2897350">I</span> <span m="2897480">will</span> <span m="2897680">use</span> <span m="2900270">a</span> <span m="2900360">little bit</span> <span m="2900750">different</span> <span m="2903113">color.</span> </p>
<p></p>
<p><span m="2913930">Let's</span> <span m="2914460">use</span> <span m="2915550">red</span> <span m="2916060">over</span> <span m="2916350">here.</span> </p>
<p></p>
<p><span m="2920230">I</span> <span m="2920330">hope</span> <span m="2920380">this</span> <span m="2920560">is</span> <span m="2920710">visible.</span> </p>
<p></p>
<p><span m="2923280">So</span> <span m="2923470">we</span> <span m="2923590">have</span> <span m="2923810">red,</span> <span m="2925740">this</span> <span m="2926000">one,</span> <span m="2927020">and</span> <span m="2927300">this</span> <span m="2927680">one,</span> <span m="2929828">and</span> <span m="2930260">this</span> <span m="2930480">one,</span> <span m="2931235">and</span> <span m="2931560">this</span> <span m="2931810">one,</span> <span m="2932390">and</span> <span m="2932830">this</span> <span m="2933120">one.</span> <span m="2934300">That's</span> <span m="2934540">going</span> <span m="2934760">to</span> <span m="2934910">be--</span> <span m="2936060">this</span> <span m="2936190">is</span> <span m="2936330">going</span> <span m="2936480">to</span> <span m="2936580">be</span> <span m="2936750">the</span> <span m="2936850">spanning</span> <span m="2937250">tree</span> <span m="2937640">that I'm</span> <span m="2937720">going</span> <span m="2938040">to</span> <span m="2938200">look</span> <span m="2939300">at.</span> <span m="2939440">This</span> <span m="2939590">is</span> <span m="2939730">also</span> <span m="2940110">weight</span> <span m="2940350">17.</span> </p>
<p></p>
<p><span m="2948750">So</span> <span m="2948910">let's</span> <span m="2949120">first</span> <span m="2949340">define</span> <span m="2949780">what</span> <span m="2949920">a</span> <span m="2950590">minimum</span> <span m="2950930">weight</span> <span m="2951130">spanning</span> <span m="2951490">tree</span> <span m="2951780">is,</span> <span m="2952390">just</span> <span m="2952690">to</span> <span m="2952820">be</span> <span m="2953160">very</span> <span m="2953830">precise.</span> <span m="2955180">And</span> <span m="2955380">then</span> <span m="2955520">let's</span> <span m="2956410">have</span> <span m="2957060">an</span> <span m="2957360">algorithm.</span> <span m="2957610">Maybe</span> <span m="2957920">you</span> <span m="2958175">have already</span> <span m="2958430">ideas</span> <span m="2959100">of</span> <span m="2959380">how</span> <span m="2959780">this</span> <span m="2959980">could</span> <span m="2960160">be</span> <span m="2960290">done.</span> </p>
<p><span m="2962790">It</span> <span m="2962960">turns</span> <span m="2963320">out</span> <span m="2963540">actually</span> <span m="2963900">that</span> <span m="2964060">if</span> <span m="2964240">you</span> <span m="2964340">just</span> <span m="2964610">start</span> <span m="2965200">with</span> <span m="2965390">a</span> <span m="2965430">greedy</span> <span m="2965730">approach,</span> <span m="2966460">you</span> <span m="2966550">just</span> <span m="2966890">start</span> <span m="2967270">to</span> <span m="2967430">add</span> <span m="2967830">sort</span> <span m="2968090">of</span> <span m="2968250">through</span> <span m="2968630">the</span> <span m="2969180">edges</span> <span m="2969540">of</span> <span m="2969650">minimum</span> <span m="2970280">weight,</span> <span m="2971420">that</span> <span m="2971570">you</span> <span m="2971660">can</span> <span m="2971840">still</span> <span m="2972280">add</span> <span m="2973350">without</span> <span m="2974600">creating</span> <span m="2975010">any</span> <span m="2975160">cycles.</span> <span m="2976630">You</span> <span m="2976820">will</span> <span m="2977010">get</span> <span m="2977090">an</span> <span m="2977220">algorithm</span> <span m="2977960">that</span> <span m="2978110">is</span> <span m="2978300">going</span> <span m="2978510">to</span> <span m="2978870">do the trick</span> <span m="2979120">for</span> <span m="2979300">you.</span> <span m="2981030">So</span> <span m="2981220">that's</span> <span m="2981450">kind</span> <span m="2981660">of</span> <span m="2981720">surprising.</span> </p>
<p><span m="2983830">So</span> <span m="2984140">let's</span> <span m="2984740">talk</span> <span m="2985130">about</span> <span m="2986110">the</span> <span m="2986200">definition.</span> </p>
<p></p>
<p><span m="2990560">So</span> <span m="2990750">the</span> <span m="2990870">minimum</span> <span m="2991270">spanning</span> <span m="2991780">tree</span> <span m="2997180">of</span> <span m="2997420">an</span> <span m="2997835">edge</span> <span m="2998250">weighted</span> <span m="2998660">graph</span> <span m="3001830">is</span> <span m="3002030">defined</span> <span m="3002130">as--</span> </p>
<p></p>
<p><span m="3012410">It's</span> <span m="3012590">defined</span> <span m="3012650">as the</span> <span m="3013100">spanning</span> <span m="3013550">tree</span> <span m="3014580">of</span> <span m="3014820">G</span> <span m="3016830">such</span> <span m="3017270">that</span> <span m="3017530">it</span> <span m="3017770">has</span> <span m="3019640">the</span> <span m="3019770">smallest</span> <span m="3023400">possible</span> <span m="3024060">sum</span> <span m="3031390">of</span> <span m="3031550">edge</span> <span m="3031770">weights.</span> </p>
<p></p>
<p><span m="3040290">Well</span> <span m="3040420">the</span> <span m="3040660">algorithm</span> <span m="3041610">that</span> <span m="3041900">I'm</span> <span m="3042080">thinking</span> <span m="3042480">about</span> <span m="3042810">here</span> <span m="3043930">is</span> <span m="3044560">very</span> <span m="3044740">straightforward.</span> <span m="3046530">Actually</span> <span m="3046865">I'll</span> <span m="3047200">use</span> <span m="3048010">this</span> <span m="3048890">blackboard</span> <span m="3049610">because</span> <span m="3050650">this</span> <span m="3050820">algorithm</span> <span m="3051310">will</span> <span m="3051440">be</span> <span m="3051590">the</span> <span m="3051700">focus</span> <span m="3052160">of</span> <span m="3052330">the</span> <span m="3052615">rest</span> <span m="3052900">of</span> <span m="3053205">the</span> <span m="3053510">lecture.</span> </p>
<p></p>
<p><span m="3068820">So</span> <span m="3069090">what's</span> <span m="3069480">the</span> <span m="3069640">algorithm?</span> </p>
<p></p>
<p><span m="3073950">We</span> <span m="3074020">actually</span> <span m="3074440">grow</span> <span m="3076070">a</span> <span m="3076170">subgraph</span> <span m="3077230">step</span> <span m="3077610">by</span> <span m="3077760">step.</span> <span m="3080430">So</span> <span m="3080720">one</span> <span m="3080970">edge</span> <span m="3081340">at</span> <span m="3081470">a</span> <span m="3081590">time</span> <span m="3090360">such</span> <span m="3090580">that</span> <span m="3090890">at</span> <span m="3091040">each</span> <span m="3091300">step</span> <span m="3095360">we</span> <span m="3095540">want</span> <span m="3095740">to</span> <span m="3095960">add</span> <span m="3101950">the</span> <span m="3102070">minimum</span> <span m="3102560">weight</span> <span m="3103180">edge</span> <span m="3115610">that</span> <span m="3115850">keeps</span> <span m="3116160">the</span> <span m="3116230">subgraph</span> <span m="3117470">acyclic.</span> </p>
<p></p>
<p><span m="3133320">So</span> <span m="3133490">this</span> <span m="3133680">is</span> <span m="3133830">our</span> <span m="3134010">algorithm.</span> <span m="3134940">Let's</span> <span m="3135130">have</span> <span m="3135290">a</span> <span m="3135350">look</span> <span m="3135790">how</span> <span m="3135980">this</span> <span m="3136210">works</span> <span m="3136930">in</span> <span m="3137060">this</span> <span m="3137240">particular</span> <span m="3137640">graph</span> <span m="3138010">over</span> <span m="3138240">here.</span> </p>
<p></p>
<p><span m="3144660">So</span> <span m="3145220">suppose</span> <span m="3146010">I</span> <span m="3146150">would</span> <span m="3146360">start</span> <span m="3146740">off</span> <span m="3147450">with--</span> <span m="3148310">well,</span> <span m="3148530">if</span> <span m="3149650">I</span> <span m="3149700">start</span> <span m="3149970">with this</span> <span m="3150200">algorithm.</span> <span m="3151080">I</span> <span m="3151190">take,</span> <span m="3151580">say,</span> <span m="3152830">any</span> <span m="3153220">minimum</span> <span m="3153510">weight</span> <span m="3153840">edge.</span> <span m="3154340">Say I</span> <span m="3154640">take</span> <span m="3154900">this</span> <span m="3155070">particular</span> <span m="3155520">one.</span> <span m="3156120">So</span> <span m="3156190">this</span> <span m="3156390">could</span> <span m="3156530">be</span> <span m="3156690">my</span> <span m="3156840">first</span> <span m="3157330">edge</span> <span m="3157600">that I</span> <span m="3157870">choose.</span> </p>
<p><span m="3160710">Then</span> <span m="3160940">I</span> <span m="3161000">may</span> <span m="3161670">want</span> <span m="3161910">to</span> <span m="3162030">choose</span> <span m="3162420">another</span> <span m="3163200">minimum</span> <span m="3163650">weight</span> <span m="3163980">edge</span> <span m="3164280">such</span> <span m="3164560">that</span> <span m="3164840">I</span> <span m="3164890">do</span> <span m="3165020">not</span> <span m="3165240">create</span> <span m="3165850">a</span> <span m="3165960">cycle.</span> <span m="3167290">Well,</span> <span m="3167530">which</span> <span m="3167720">one</span> <span m="3167890">could</span> <span m="3168080">I</span> <span m="3168140">choose?</span> <span m="3168490">I</span> <span m="3168600">could</span> <span m="3168740">choose</span> <span m="3169050">this</span> <span m="3169270">one.</span> <span m="3169770">I</span> <span m="3169860">could</span> <span m="3170070">also</span> <span m="3170280">choose</span> <span m="3170590">this</span> <span m="3170996">one.</span> <span m="3172620">With</span> <span m="3172800">this one,</span> <span m="3172860">I could</span> <span m="3173070">choose</span> <span m="3173340">this</span> <span m="3173650">part.</span> <span m="3174160">With</span> <span m="3174320">this</span> <span m="3174520">one,</span> <span m="3174650">I</span> <span m="3174690">could</span> <span m="3174890">choose</span> <span m="3175100">that</span> <span m="3175330">part.</span> <span m="3175770">I</span> <span m="3175840">could</span> <span m="3176020">also</span> <span m="3176240">choose</span> <span m="3176520">this</span> <span m="3176740">one.</span> </p>
<p><span m="3177430">So</span> <span m="3177580">let's</span> <span m="3178010">choose</span> <span m="3178310">this</span> <span m="3178520">one</span> <span m="3178720">actually.</span> <span m="3180780">This</span> <span m="3180970">would</span> <span m="3181100">be our</span> <span m="3181260">second</span> <span m="3182300">step</span> <span m="3182870">and</span> <span m="3183270">our</span> <span m="3183960">second</span> <span m="3184390">edge that we</span> <span m="3184840">select</span> <span m="3185210">in</span> <span m="3185300">our</span> <span m="3185670">algorithm.</span> <span m="3186490">There</span> <span m="3186590">are</span> <span m="3186830">two</span> <span m="3187010">disjoint</span> <span m="3187680">edges.</span> <span m="3188130">So</span> <span m="3188510">there's</span> <span m="3188700">definitely</span> <span m="3189120">no</span> <span m="3189320">graph.</span> </p>
<p><span m="3192600">Well,</span> <span m="3193150">maybe</span> <span m="3193420">now</span> <span m="3193670">our</span> <span m="3193950">third</span> <span m="3194230">step,</span> <span m="3195660">I</span> <span m="3195810">can</span> <span m="3196000">choose</span> <span m="3196290">one</span> <span m="3196470">of</span> <span m="3196630">these</span> <span m="3197790">edges,</span> <span m="3199470">which</span> <span m="3200120">also</span> <span m="3200480">has</span> <span m="3200780">just</span> <span m="3201120">weight</span> <span m="3201390">1,</span> <span m="3201670">which</span> <span m="3201810">is</span> <span m="3201930">minimum</span> <span m="3202270">weight.</span> <span m="3203030">So</span> <span m="3203530">this</span> <span m="3203690">could</span> <span m="3203810">be</span> <span m="3203940">my</span> <span m="3204120">third</span> <span m="3204960">possibility.</span> </p>
<p><span m="3207960">Then</span> <span m="3208110">what's</span> <span m="3208360">the</span> <span m="3208450">fourth</span> <span m="3208840">possibility?</span> <span m="3209570">Well,</span> <span m="3209840">I</span> <span m="3210060">have</span> <span m="3212830">to</span> <span m="3212970">choose</span> <span m="3213380">one</span> <span m="3213600">of</span> <span m="3213740">those</span> <span m="3214290">actually.</span> <span m="3215890">I</span> <span m="3215980">cannot</span> <span m="3216290">choose</span> <span m="3219480">this</span> <span m="3219680">weight</span> <span m="3219920">1</span> <span m="3220160">edge?</span> <span m="3220410">Why</span> <span m="3220590">not?</span> <span m="3220850">Because</span> <span m="3221160">that</span> <span m="3221370">would</span> <span m="3221530">create</span> <span m="3221830">a</span> <span m="3221890">cycle.</span> </p>
<p><span m="3223400">So</span> <span m="3223560">I'm</span> <span m="3223660">not</span> <span m="3223860">allowed</span> <span m="3224070">to</span> <span m="3224160">choose</span> <span m="3224410">this</span> <span m="3224590">one</span> <span m="3224750">anymore.</span> <span m="3225520">So I</span> <span m="3225780">choose</span> <span m="3226060">one</span> <span m="3226240">of</span> <span m="3226350">these</span> <span m="3226640">two.</span> </p>
<p><span m="3227340">So</span> <span m="3227520">for</span> <span m="3227700">example,</span> <span m="3228210">I</span> <span m="3228290">choose</span> <span m="3229870">this</span> <span m="3230230">one</span> <span m="3230410">over</span> <span m="3230660">here.</span> <span m="3230860">This is</span> <span m="3231020">the</span> <span m="3231150">fourth</span> <span m="3232990">edge</span> <span m="3233150">that I</span> <span m="3233470">add</span> <span m="3233710">to</span> <span m="3233830">the</span> <span m="3233950">whole</span> <span m="3234440">picture.</span> </p>
<p><span m="3235410">So</span> <span m="3235790">note</span> <span m="3235940">that</span> <span m="3236120">in</span> <span m="3236420">the</span> <span m="3236700">meantime.</span> <span m="3237200">I</span> <span m="3237330">have</span> <span m="3237520">two</span> <span m="3237720">disconnected</span> <span m="3238480">components.</span> <span m="3240370">One</span> <span m="3240757">other</span> <span m="3241531">here,</span> <span m="3242360">which</span> <span m="3242690">is</span> <span m="3242860">this</span> <span m="3243160">arc,</span> <span m="3244150">and</span> <span m="3244380">another</span> <span m="3244600">arc</span> <span m="3245070">over</span> <span m="3245190">here.</span> </p>
<p><span m="3247300">So</span> <span m="3248070">which</span> <span m="3248590">edge</span> <span m="3248900">do</span> <span m="3249205">I</span> <span m="3249510">attach</span> <span m="3249970">now</span> <span m="3250530">in</span> <span m="3250670">the</span> <span m="3250800">algorithm?</span> <span m="3251860">Well,</span> <span m="3252680">I</span> <span m="3252770">can</span> <span m="3252960">still</span> <span m="3253580">add</span> <span m="3254000">this</span> <span m="3254290">minimum</span> <span m="3254580">weight</span> <span m="3254870">edge</span> <span m="3255050">over</span> <span m="3255420">here</span> <span m="3255770">of</span> <span m="3256400">weight</span> <span m="3256810">two,</span> <span m="3257780">because</span> <span m="3258110">it</span> <span m="3258180">does</span> <span m="3258340">not</span> <span m="3258810">create</span> <span m="3259120">any</span> <span m="3259350">cycle.</span> <span m="3260140">So</span> <span m="3260330">this</span> <span m="3260530">would</span> <span m="3260660">be</span> <span m="3261710">the</span> <span m="3261810">fifth</span> <span m="3262490">wonder</span> <span m="3262860">that I</span> <span m="3263100">add</span> <span m="3263340">to</span> <span m="3263450">the</span> <span m="3263540">whole</span> <span m="3263720">picture.</span> </p>
<p><span m="3265500">Well,</span> <span m="3265790">now</span> <span m="3266030">I</span> <span m="3266140">can</span> <span m="3267060">choose</span> <span m="3267520">either</span> <span m="3267820">one</span> <span m="3267990">of</span> <span m="3268130">those.</span> <span m="3268750">So</span> <span m="3269130">for</span> <span m="3269480">example,</span> <span m="3269680">this</span> <span m="3269860">would</span> <span m="3269980">be</span> <span m="3270120">my</span> <span m="3270260">sixth</span> <span m="3270620">step.</span> <span m="3271310">And</span> <span m="3271430">then</span> <span m="3271560">finally,</span> <span m="3272440">I</span> <span m="3272570">can</span> <span m="3272800">choose</span> <span m="3273150">none</span> <span m="3273850">of</span> <span m="3274200">these</span> <span m="3275420">because</span> <span m="3276030">those</span> <span m="3276270">would</span> <span m="3276420">create</span> <span m="3276760">cycles.</span> <span m="3277410">But</span> <span m="3277550">I</span> <span m="3277600">can</span> <span m="3277960">choose</span> <span m="3278290">this</span> <span m="3278490">one.</span> <span m="3279350">So</span> <span m="3279550">this</span> <span m="3279760">is</span> <span m="3279880">going</span> <span m="3280100">to</span> <span m="3280230">be</span> <span m="3281250">the</span> <span m="3281350">seventh</span> <span m="3281860">one</span> <span m="3282440">that</span> <span m="3282860">I add</span> <span m="3283120">to</span> <span m="3283310">the</span> <span m="3283420">whole</span> <span m="3285660">graph</span> <span m="3286030">that</span> <span m="3286400">I</span> <span m="3286470">construct</span> <span m="3287080">here</span> <span m="3287270">and</span> <span m="3287330">grow</span> <span m="3287740">in</span> <span m="3287870">this</span> <span m="3288050">algorithm.</span> </p>
<p><span m="3289060">It</span> <span m="3289260">turns</span> <span m="3289640">out that</span> <span m="3289690">this</span> <span m="3289850">algorithm</span> <span m="3290300">actually</span> <span m="3290670">creates</span> <span m="3291130">for</span> <span m="3291260">you</span> <span m="3291510">a</span> <span m="3291580">minimum</span> <span m="3291870">spanning</span> <span m="3292280">tree.</span> <span m="3293360">And</span> <span m="3293520">the</span> <span m="3293600">proof</span> <span m="3293910">is</span> <span m="3294700">much</span> <span m="3294930">more</span> <span m="3295110">complex</span> <span m="3295700">than</span> <span m="3295840">what</span> <span m="3295980">you have</span> <span m="3296160">seen</span> <span m="3296420">before.</span> <span m="3298120">So</span> <span m="3298250">that's</span> <span m="3299600">what</span> <span m="3299770">we're</span> <span m="3299900">going</span> <span m="3300110">to</span> <span m="3300230">do</span> <span m="3300430">right</span> <span m="3300670">now.</span> </p>
<p><span m="3303650">So</span> <span m="3303850">when</span> <span m="3304020">we</span> <span m="3304170">think</span> <span m="3304390">about</span> <span m="3304620">this</span> <span m="3304760">algorithm,</span> <span m="3306050">then</span> <span m="3306720">somehow</span> <span m="3307180">I</span> <span m="3307290">want</span> <span m="3307490">to</span> <span m="3307590">guarantee</span> <span m="3308030">that</span> <span m="3308440">at</span> <span m="3308780">every</span> <span m="3309120">single</span> <span m="3309480">step,</span> <span m="3310220">I</span> <span m="3310340">will</span> <span m="3310560">be</span> <span m="3310770">able</span> <span m="3311130">to</span> <span m="3311280">grow</span> <span m="3311720">into</span> <span m="3312190">a</span> <span m="3312230">minimum</span> <span m="3312590">spanning</span> <span m="3313020">tree.</span> <span m="3314090">And</span> <span m="3314910">that's</span> <span m="3315760">the</span> <span m="3315910">basic</span> <span m="3316970">intuition</span> <span m="3317570">that</span> <span m="3317680">we</span> <span m="3317770">have.</span> <span m="3318090">And  that's the</span> <span m="3318260">lemma</span> <span m="3318630">that</span> <span m="3318860">we</span> <span m="3318900">want</span> <span m="3319080">to</span> <span m="3319170">prove</span> <span m="3319410">first.</span> </p>
<p><span m="3321520">So</span> <span m="3322670">formalizing,</span> <span m="3325415">we</span> <span m="3325860">get</span> <span m="3327960">the</span> <span m="3328060">following.</span> <span m="3329520">We want</span> <span m="3329760">to</span> <span m="3330060">prove the</span> <span m="3330150">following</span> <span m="3330510">statement.</span> </p>
<p></p>
<p><span m="3339040">So</span> <span m="3339280">the</span> <span m="3339380">lemma</span> <span m="3340090">that will</span> <span m="3340420">help</span> <span m="3340780">us</span> <span m="3342340">is</span> <span m="3343580">let</span> <span m="3344040">S</span> <span m="3345390">be</span> <span m="3345860">the</span> <span m="3345930">set</span> <span m="3346220">of</span> <span m="3346400">edges</span> <span m="3346870">that</span> <span m="3347150">I</span> <span m="3347290">have</span> <span m="3347620">been</span> <span m="3347760">selecting</span> <span m="3349270">up</span> <span m="3349500">to</span> <span m="3352230">now</span> <span m="3352470">in</span> <span m="3352540">the</span> <span m="3352660">algorithm.</span> <span m="3353100">So let</span> <span m="3353480">S</span> <span m="3353800">consist,</span> <span m="3354950">for</span> <span m="3355120">example,</span> <span m="3356390">of</span> <span m="3357150">the</span> <span m="3357270">first</span> <span m="3358720">m</span> <span m="3359690">edges.</span> </p>
<p></p>
<p><span m="3370250">What</span> <span m="3370440">we</span> <span m="3370580">want</span> <span m="3370730">to</span> <span m="3370820">show</span> <span m="3371260">is</span> <span m="3375188">that</span> <span m="3377050">we</span> <span m="3377270">can</span> <span m="3377430">still</span> <span m="3377760">extend</span> <span m="3378530">this</span> <span m="3378730">set</span> <span m="3378910">of</span> <span m="3379090">edges</span> <span m="3379530">into</span> <span m="3379790">a</span> <span m="3379950">minimum</span> <span m="3380190">spanning</span> <span m="3380590">tree.</span> <span m="3380960">We</span> <span m="3381040">can</span> <span m="3381180">still</span> <span m="3381430">grow</span> <span m="3381930">within</span> <span m="3382310">the</span> <span m="3382440">algorithm</span> <span m="3383190">into</span> <span m="3383850">a</span> <span m="3383980">minimum</span> <span m="3384270">spanning</span> <span m="3384670">tree.</span> <span m="3384950">That</span> <span m="3385190">is</span> <span m="3385350">what</span> <span m="3385480">you</span> <span m="3385620">want</span> <span m="3385760">to</span> <span m="3385830">show.</span> </p>
<p><span m="3388280">So</span> <span m="3388640">we want to</span> <span m="3388740">show</span> <span m="3389160">that</span> <span m="3389350">their</span> <span m="3389700">exists</span> <span m="3390750">a</span> <span m="3391820">minimum</span> <span m="3392240">spanning</span> <span m="3392700">tree</span> <span m="3394300">T</span> <span m="3395150">that</span> <span m="3395370">has</span> <span m="3396110">the</span> <span m="3397050">vertex</span> <span m="3397390">set</span> <span m="3397780">V</span> <span m="3398410">and</span> <span m="3398890">an</span> <span m="3399060">edge set</span> <span m="3399380">E.</span> </p>
<p></p>
<p><span m="3403270">So</span> <span m="3403450">this</span> <span m="3403680">is the</span> <span m="3403720">minimum</span> <span m="3403930">spanning</span> <span m="3404240">tree</span> <span m="3404510">for</span> <span m="3405150">the</span> <span m="3405290">graph</span> <span m="3405650">G</span> <span m="3406220">such</span> <span m="3406720">that</span> <span m="3413720">S</span> <span m="3414530">is</span> <span m="3414780">actually</span> <span m="3415540">a</span> <span m="3415640">subset</span> <span m="3416460">of</span> <span m="3417140">the</span> <span m="3417340">edges</span> <span m="3418000">in</span> <span m="3418190">this</span> <span m="3418370">minimum</span> <span m="3418650">spanning</span> <span m="3419030">tree.</span> <span m="3419200">So</span> <span m="3419710">this</span> <span m="3419920">is</span> <span m="3420470">a</span> <span m="3420530">nice</span> <span m="3420870">mathematical</span> <span m="3421430">formulation</span> <span m="3422780">that</span> <span m="3424070">really</span> <span m="3425800">precisely</span> <span m="3426530">states</span> <span m="3427280">that</span> <span m="3427860">we</span> <span m="3428020">can</span> <span m="3428190">still</span> <span m="3428900">keep</span> <span m="3429190">on</span> <span m="3429310">growing</span> <span m="3429740">into</span> <span m="3430030">a</span> <span m="3430080">minimum</span> <span m="3430400">spanning</span> <span m="3430610">tree.</span> </p>
<p><span m="3430710">So</span> <span m="3430950">how</span> <span m="3431240">would</span> <span m="3431480">this</span> <span m="3431620">lemma</span> <span m="3431750">help</span> <span m="3432040">us</span> <span m="3432140">in</span> <span m="3432500">proving</span> <span m="3433100">the</span> <span m="3433190">theorem</span> <span m="3433560">that</span> <span m="3433710">we</span> <span m="3433950">want</span> <span m="3434160">to</span> <span m="3434600">show?</span> <span m="3436980">Actually,</span> <span m="3437300">where</span> <span m="3437670">is the</span> <span m="3437730">theorem?</span> <span m="3440110">I</span> <span m="3440260">did</span> <span m="3440420">not</span> <span m="3440710">write</span> <span m="3440920">down</span> <span m="3441140">the</span> <span m="3441220">theorem</span> <span m="3441690">so</span> <span m="3441950">far.</span> <span m="3443080">I can</span> <span m="3443455">put</span> <span m="3443830">this</span> <span m="3444240">over</span> <span m="3444560">here.</span> </p>
<p><span m="3445180">So</span> <span m="3445390">the</span> <span m="3445620">theorem</span> <span m="3445980">that's</span> <span m="3446160">want</span> <span m="3446420">to</span> <span m="3446510">show</span> <span m="3446910">is</span> <span m="3449740">that</span> <span m="3450500">for</span> <span m="3450950">any</span> <span m="3454107">connected</span> <span m="3458800">weighted</span> <span m="3459290">graph</span> <span m="3459730">G,</span> <span m="3463990">the</span> <span m="3464160">algorithm</span> <span m="3464880">creates</span> <span m="3465250">a minimum</span> <span m="3465530">spanning</span> <span m="3465900">tree.</span> </p>
<p></p>
<p><span m="3471990">The algorithm</span> <span m="3472320">produces</span> <span m="3477530">a</span> <span m="3477720">minimum</span> <span m="3477950">spanning</span> <span m="3478340">tree.</span> </p>
<p><span m="3479960">So</span> <span m="3480120">how</span> <span m="3480310">can</span> <span m="3480440">we</span> <span m="3480550">prove</span> <span m="3480790">his</span> <span m="3481020">theorem?</span> <span m="3482580">Suppose</span> <span m="3483090">we</span> <span m="3483200">know</span> <span m="3483580">that this</span> <span m="3483670">lemma</span> <span m="3484210">is</span> <span m="3484525">true,</span> <span m="3484840">which</span> <span m="3485180">we</span> <span m="3485250">will</span> <span m="3485480">prove</span> <span m="3485770">later.</span> <span m="3487050">Then</span> <span m="3488420">how</span> <span m="3488550">do</span> <span m="3488640">we</span> <span m="3488780">know</span> <span m="3489060">that</span> <span m="3489340">this</span> <span m="3489440">theorem</span> <span m="3489725">holds?</span> </p>
<p></p>
<p><span m="3496020">Actually,</span> <span m="3496375">I'm</span> <span m="3496730">wiping</span> <span m="3497120">out</span> <span m="3497240">this</span> <span m="3497370">particular</span> <span m="3497740">theorem</span> <span m="3498005">over</span> <span m="3498270">here.</span> <span m="3498630">We will</span> <span m="3498780">use</span> <span m="3499100">this</span> <span m="3499260">in</span> <span m="3499350">a</span> <span m="3499390">moment.</span> </p>
<p><span m="3500530">So</span> <span m="3500830">we</span> <span m="3500950">know</span> <span m="3501180">that</span> <span m="3501490">every</span> <span m="3501820">connected</span> <span m="3502470">graph</span> <span m="3503060">has</span> <span m="3503260">a</span> <span m="3503360">spanning</span> <span m="3503740">tree.</span> <span m="3503990">We</span> <span m="3504110">already</span> <span m="3504400">know</span> <span m="3504640">that.</span> <span m="3506740">But</span> <span m="3506900">now</span> <span m="3507040">we</span> <span m="3507150">want</span> <span m="3507300">to</span> <span m="3507390">show</span> <span m="3507730">that</span> <span m="3508370">we</span> <span m="3508470">can</span> <span m="3508640">even</span> <span m="3508880">construct</span> <span m="3509030">a</span> <span m="3509420">minimum</span> <span m="3509680">spanning</span> <span m="3510060">tree</span> <span m="3510250">for</span> <span m="3510740">a</span> <span m="3510860">weighted</span> <span m="3511210">graph.</span> </p>
<p></p>
<p><span m="3514070">So</span> <span m="3514950">how</span> <span m="3515170">are</span> <span m="3515280">we</span> <span m="3515390">going</span> <span m="3515740">ahead?</span> <span m="3518850">The</span> <span m="3518980">proof</span> <span m="3519390">of the</span> <span m="3519560">theorem</span> <span m="3524360">is</span> <span m="3524600">as</span> <span m="3524740">follows.</span> <span m="3525820">I</span> <span m="3525840">will</span> <span m="3526000">suppose</span> <span m="3526410">that</span> <span m="3526530">the</span> <span m="3526610">number</span> <span m="3526940">of</span> <span m="3527280">edges--</span> <span m="3527940">the</span> <span m="3528150">number</span> <span m="3528290">of</span> <span m="3528590">vertices--</span> <span m="3529820">is</span> <span m="3530040">actually</span> <span m="3530380">equal</span> <span m="3530730">to</span> <span m="3531010">n.</span> <span m="3532320">And</span> <span m="3534290">if</span> <span m="3534530">we</span> <span m="3534700">want</span> <span m="3534950">to</span> <span m="3535070">show</span> <span m="3535390">that</span> <span m="3535580">we</span> <span m="3535730">get</span> <span m="3536020">to</span> <span m="3536180">a</span> <span m="3536300">minimum</span> <span m="3536540">spanning</span> <span m="3536960">tree,</span> <span m="3537750">we</span> <span m="3537870">want</span> <span m="3538020">to</span> <span m="3538120">first</span> <span m="3538420">of</span> <span m="3538520">all</span> <span m="3538670">show</span> <span m="3539050">that</span> <span m="3539270">the</span> <span m="3539730">algorithm</span> <span m="3540820">is</span> <span m="3541090">able</span> <span m="3541410">to</span> <span m="3541570">choose</span> <span m="3542540">at</span> <span m="3542950">least</span> <span m="3543690">n</span> <span m="3543930">minus</span> <span m="3544310">1</span> <span m="3544580">edges,</span> <span m="3545550">right?</span> <span m="3545840">Because</span> <span m="3546160">a</span> <span m="3546290">tree</span> <span m="3546970">has</span> <span m="3547290">n</span> <span m="3547450">minus</span> <span m="3547780">1</span> <span m="3547940">edges.</span> <span m="3548100">We</span> <span m="3548340">have</span> <span m="3548650">shown</span> <span m="3548950">this</span> <span m="3549120">before.</span> </p>
<p></p>
<p><span m="3553220">The</span> <span m="3553430">algorithm</span> <span m="3553960">should</span> <span m="3554180">not</span> <span m="3554380">get</span> <span m="3554550">stuck.</span> <span m="3555030">So</span> <span m="3555200">let's</span> <span m="3555420">first</span> <span m="3555660">prove</span> <span m="3555940">that.</span> </p>
<p><span m="3557280">So</span> <span m="3557900">suppose</span> <span m="3559350">I</span> <span m="3559550">have</span> <span m="3560100">chosen</span> <span m="3560680">less</span> <span m="3561390">than</span> <span m="3561660">n</span> <span m="3561880">minus</span> <span m="3562260">1</span> <span m="3562670">edges.</span> <span m="3566310">So</span> <span m="3566540">suppose</span> <span m="3570190">we have</span> <span m="3570440">less</span> <span m="3570690">than</span> <span m="3570820">n</span> <span m="3570950">minus</span> <span m="3571180">1</span> <span m="3571280">edges</span> <span m="3571710">picked.</span> <span m="3571920">Well</span> <span m="3572080">then</span> <span m="3573130">what</span> <span m="3573280">do</span> <span m="3573370">we</span> <span m="3573490">know?</span> </p>
<p><span m="3575220">Well,</span> <span m="3575930">there</span> <span m="3576240">exists</span> <span m="3576550">as</span> <span m="3576860">edge</span> <span m="3580510">in</span> <span m="3581700">E</span> <span m="3582070">minus</span> <span m="3582650">S--</span> <span m="3584460">well,</span> <span m="3585120">let</span> <span m="3585190">me</span> <span m="3585500">first</span> <span m="3585780">write</span> <span m="3585960">it</span> <span m="3586090">out--</span> <span m="3588510">that</span> <span m="3588720">can</span> <span m="3588920">be</span> <span m="3589140">added</span> <span m="3594970">without</span> <span m="3596660">creating</span> <span m="3597600">a</span> <span m="3598060">cycle.</span> </p>
<p></p>
<p><span m="3602120">So</span> <span m="3602360">why</span> <span m="3602540">is</span> <span m="3602750">this?</span> <span m="3603760">Well,</span> <span m="3604810">I</span> <span m="3604900">know</span> <span m="3605150">that</span> <span m="3605490">E</span> <span m="3607240">is</span> <span m="3607600">the edge</span> <span m="3607760">set</span> <span m="3608220">of</span> <span m="3608500">my</span> <span m="3609100">minimum</span> <span m="3609440">spanning</span> <span m="3609860">tree.</span> <span m="3610760">I</span> <span m="3610890">know</span> <span m="3611310">that</span> <span m="3611600">S is a</span> <span m="3611840">part</span> <span m="3612760">of--</span> <span m="3615970">is</span> <span m="3616480">a</span> <span m="3616530">subset</span> <span m="3616910">of all</span> <span m="3617170">the</span> <span m="3617300">edges.</span> <span m="3618390">I</span> <span m="3618530">know</span> <span m="3618790">that</span> <span m="3618980">a</span> <span m="3619030">minimum</span> <span m="3619290">spanning</span> <span m="3619750">tree</span> <span m="3620020">is</span> <span m="3620180">a</span> <span m="3620280">tree.</span> <span m="3621490">It's</span> <span m="3621750">a</span> <span m="3621860">tree</span> <span m="3622160">on</span> <span m="3622390">n</span> <span m="3622640">vertices.</span> </p>
<p><span m="3624570">We</span> <span m="3624700">know</span> <span m="3624940">that's</span> <span m="3625340">a</span> <span m="3625430">tree</span> <span m="3625770">on</span> <span m="3626110">n vertices</span> <span m="3626770">has</span> <span m="3627120">n</span> <span m="3627310">minus</span> <span m="3627720">1</span> <span m="3628320">edges.</span> <span m="3629870">I</span> <span m="3630000">know</span> <span m="3630250">that</span> <span m="3631070">less</span> <span m="3631400">than</span> <span m="3631570">n</span> <span m="3631720">minus</span> <span m="3632020">1</span> <span m="3632170">edges</span> <span m="3632570">have</span> <span m="3632760">been</span> <span m="3633220">chosen</span> <span m="3633650">so</span> <span m="3633910">far.</span> <span m="3635070">So</span> <span m="3635240">that</span> <span m="3635430">means</span> <span m="3635770">that</span> <span m="3636310">S</span> <span m="3636920">has</span> <span m="3637160">less</span> <span m="3637520">than</span> <span m="3637730">n</span> <span m="3637890">minus</span> <span m="3638290">1</span> <span m="3639350">edges.</span> <span m="3640720">E</span> <span m="3640930">has</span> <span m="3641070">successfully</span> <span m="3641680">n</span> <span m="3641800">minus</span> <span m="3642140">1</span> <span m="3642290">edges.</span> <span m="3642650">So</span> <span m="3642780">there's</span> <span m="3643000">at</span> <span m="3643110">least</span> <span m="3643870">one</span> <span m="3644180">edge</span> <span m="3645030">that</span> <span m="3645230">is</span> <span m="3645570">contained,</span> <span m="3645915">that</span> <span m="3646260">it</span> <span m="3646650">is</span> <span m="3646760">an</span> <span m="3646910">element</span> <span m="3647580">in</span> <span m="3647780">E,</span> <span m="3648810">but</span> <span m="3648970">not</span> <span m="3649170">in</span> <span m="3649250">S.</span> </p>
<p><span m="3652345">And</span> <span m="3652790">for</span> <span m="3652990">that</span> <span m="3653200">particular</span> <span m="3653820">edge--</span> <span m="3655420">well,</span> <span m="3655610">since</span> <span m="3655920">it's</span> <span m="3656110">part</span> <span m="3656350">of</span> <span m="3656470">this</span> <span m="3656865">tree,</span> <span m="3657910">together</span> <span m="3658390">with</span> <span m="3660240">S,</span> <span m="3660520">which</span> <span m="3660690">is</span> <span m="3660810">also</span> <span m="3661060">part</span> <span m="3661550">of</span> <span m="3663280">all</span> <span m="3663740">the</span> <span m="3663950">edges</span> <span m="3664160">within</span> <span m="3664290">the tree--</span> <span m="3665340">well,</span> <span m="3665680">we</span> <span m="3665800">know</span> <span m="3666670">that</span> <span m="3666930">that</span> <span m="3667170">edge,</span> <span m="3667950">for</span> <span m="3668200">that</span> <span m="3668410">reason,</span> <span m="3668770">will</span> <span m="3668890">not</span> <span m="3669100">create</span> <span m="3669340">a</span> <span m="3669560">cycle.</span> <span m="3671990">Because</span> <span m="3672390">otherwise,</span> <span m="3673150">the</span> <span m="3673380">original</span> <span m="3674130">would</span> <span m="3674270">have</span> <span m="3674470">a</span> <span m="3674520">cycle.</span> </p>
<p><span m="3677690">So</span> <span m="3677920">we</span> <span m="3678040">know</span> <span m="3678230">that</span> <span m="3678400">the</span> <span m="3678590">algorithm</span> <span m="3679070">can</span> <span m="3679240">always</span> <span m="3680340">select</span> <span m="3681080">an</span> <span m="3681290">edge</span> <span m="3682560">such</span> <span m="3682890">that's</span> <span m="3683120">no</span> <span m="3683460">cycle</span> <span m="3683940">is</span> <span m="3684550">created.</span> <span m="3685300">And</span> <span m="3685470">that</span> <span m="3685650">means</span> <span m="3685950">that</span> <span m="3686080">we</span> <span m="3686640">can</span> <span m="3686820">keep</span> <span m="3687060">on</span> <span m="3687160">going</span> <span m="3687550">in</span> <span m="3687670">the</span> <span m="3687830">algorithm</span> <span m="3688960">until</span> <span m="3690410">at</span> <span m="3690550">least</span> <span m="3690930">n</span> <span m="3691100">minus</span> <span m="3691470">1</span> <span m="3691730">edges</span> <span m="3692120">are</span> <span m="3692730">selected.</span> </p>
<p><span m="3693410">So</span> <span m="3693990">what</span> <span m="3694130">happens</span> <span m="3695200">if</span> <span m="3695420">we</span> <span m="3695510">have</span> <span m="3695750">n</span> <span m="3695890">minus</span> <span m="3696220">1</span> <span m="3696380">edges?</span> </p>
<p></p>
<p><span m="3700020">So</span> <span m="3700350">once</span> <span m="3704540">n</span> <span m="3704730">minus</span> <span m="3705120">1</span> <span m="3706580">edges</span> <span m="3706800">have</span> <span m="3707230">been</span> <span m="3707410">chosen,</span> <span m="3709960">well,</span> <span m="3710410">then</span> <span m="3710750">S</span> <span m="3711500">has</span> <span m="3711830">exactly</span> <span m="3712940">n</span> <span m="3713140">minus</span> <span m="3713440">1</span> <span m="3713780">edges.</span> <span m="3714500">E</span> <span m="3714750">has</span> <span m="3714980">exactly</span> <span m="3715440">n</span> <span m="3715580">minus</span> <span m="3715930">1</span> <span m="3716150">edges,</span> <span m="3716960">because</span> <span m="3717280">it's</span> <span m="3717450">a</span> <span m="3717520">tree.</span> <span m="3718450">They are</span> <span m="3718680">subsets</span> <span m="3719200">of</span> <span m="3719320">one</span> <span m="3719490">another.</span> <span m="3719930">So</span> <span m="3720090">they</span> <span m="3720190">must</span> <span m="3720410">be</span> <span m="3720590">equal</span> <span m="3720830">to</span> <span m="3720960">one</span> <span m="3721110">another.</span> </p>
<p><span m="3722330">So</span> <span m="3722410">essentially</span> <span m="3723240">what</span> <span m="3723380">we</span> <span m="3723520">have</span> <span m="3723800">here</span> <span m="3723980">is</span> <span m="3724160">that</span> <span m="3724490">S</span> <span m="3725530">exactly</span> <span m="3726150">defines</span> <span m="3727410">the</span> <span m="3727510">minimum</span> <span m="3727860">spanning</span> <span m="3728320">tree</span> <span m="3730040">that</span> <span m="3730290">we</span> <span m="3730680">are</span> <span m="3730870">looking</span> <span m="3731170">for.</span> <span m="3731730">So</span> <span m="3732630">the</span> <span m="3732770">algorithm</span> <span m="3734270">at</span> <span m="3734420">the</span> <span m="3734490">very</span> <span m="3734820">end,</span> <span m="3735880">when</span> <span m="3736040">it</span> <span m="3736140">has</span> <span m="3736410">chosen</span> <span m="3736790">n</span> <span m="3736940">minus</span> <span m="3737220">1</span> <span m="3737360">edges,</span> <span m="3738010">produces</span> <span m="3738570">a</span> <span m="3738630">minimum</span> <span m="3738930">spanning</span> <span m="3739300">tree.</span> <span m="3741540">So</span> <span m="3741770">we</span> <span m="3741890">know</span> <span m="3745050">that</span> <span m="3745320">S</span> <span m="3747600">defines</span> <span m="3748090">the</span> <span m="3748210">edges</span> <span m="3749290">of</span> <span m="3753540">a</span> <span m="3753800">minimum</span> <span m="3753940">weight</span> <span m="3754130">spanning</span> <span m="3754570">tree.</span> <span m="3755970">So</span> <span m="3756670">if</span> <span m="3756930">this</span> <span m="3757130">lemma</span> <span m="3757460">really</span> <span m="3757720">holds</span> <span m="3758080">true,</span> <span m="3758610">then</span> <span m="3758750">this</span> <span m="3759350">theorem</span> <span m="3759830">can</span> <span m="3760070">be</span> <span m="3760200">shown.</span> </p>
<p><span m="3761300">So</span> <span m="3761390">now</span> <span m="3761550">let's</span> <span m="3761800">have</span> <span m="3761970">a</span> <span m="3762050">look</span> <span m="3762250">at</span> <span m="3762290">this</span> <span m="3762490">lemma.</span> <span m="3762850">And</span> <span m="3763010">we're</span> <span m="3763160">going</span> <span m="3763350">to</span> <span m="3763500">use</span> <span m="3763720">that</span> <span m="3763790">picture</span> <span m="3764200">over</span> <span m="3764460">there</span> <span m="3764830">to</span> <span m="3764960">see</span> <span m="3765650">how</span> <span m="3765820">we</span> <span m="3765930">can</span> <span m="3766070">prove</span> <span m="3766420">this.</span> <span m="3767610">So</span> <span m="3767690">this</span> <span m="3767890">is</span> <span m="3768060">a</span> <span m="3768120">little</span> <span m="3768370">bit</span> <span m="3769830">tricky.</span> <span m="3771610">And</span> <span m="3773210">the</span> <span m="3773340">way</span> <span m="3773500">to</span> <span m="3773640">go</span> <span m="3773850">about</span> <span m="3774230">this</span> <span m="3774870">is</span> <span m="3776690">actually</span> <span m="3777160">to</span> <span m="3777330">use</span> <span m="3777960">induction.</span> <span m="3780340">So</span> <span m="3780490">let's</span> <span m="3780910">check</span> <span m="3782490">see</span> <span m="3782670">whether</span> <span m="3782830">we</span> <span m="3782940">can</span> <span m="3783070">do</span> <span m="3783190">this</span> <span m="3783430">together.</span> </p>
<p></p>
<p><span m="3791800">So</span> <span m="3792020">the</span> <span m="3792110">proof</span> <span m="3792310">of</span> <span m="3792650">the lemma is</span> <span m="3793120">as</span> <span m="3793260">follows.</span> </p>
<p></p>
<p><span m="3800620">We</span> <span m="3800750">want</span> <span m="3800950">to</span> <span m="3801090">use</span> <span m="3802260">induction.</span> <span m="3803940">On</span> <span m="3804140">what</span> <span m="3804480">do</span> <span m="3804550">you</span> <span m="3804710">think</span> <span m="3804980">we</span> <span m="3805050">are</span> <span m="3805150">going</span> <span m="3805380">to</span> <span m="3805490">induct</span> <span m="3805890">here?</span> <span m="3807160">Well,</span> <span m="3807480">we</span> <span m="3807600">want</span> <span m="3807800">to</span> <span m="3807900">prove</span> <span m="3808240">this</span> <span m="3808380">particular</span> <span m="3808770">lemma,</span> <span m="3809560">and</span> <span m="3809810">we</span> <span m="3809930">have</span> <span m="3810150">this</span> <span m="3810300">particular</span> <span m="3810930">variable</span> <span m="3811210">m.</span> <span m="3811490">So</span> <span m="3811770">that</span> <span m="3811890">seems</span> <span m="3812100">to</span> <span m="3812190">be</span> <span m="3812370">a</span> <span m="3812420">good</span> <span m="3813160">methods</span> <span m="3813720">to</span> <span m="3814480">continue.</span> <span m="3814790">So</span> <span m="3815090">we'll</span> <span m="3815390">use</span> <span m="3815670">induction</span> <span m="3816240">on</span> <span m="3816590">m.</span> </p>
<p></p>
<p><span m="3821740">Let's</span> <span m="3822240">state</span> <span m="3822640">the</span> <span m="3824960">induction</span> <span m="3825650">hypothesis.</span> <span m="3827030">Well, it's</span> <span m="3827140">exactly</span> <span m="3827520">the</span> <span m="3827630">same as</span> <span m="3828050">in the</span> <span m="3828390">lemma.</span> <span m="3829000">So</span> <span m="3829500">we</span> <span m="3829570">say</span> <span m="3830760">for</span> <span m="3831050">all</span> <span m="3831350">G</span> <span m="3832270">and</span> <span m="3832730">for</span> <span m="3833060">all</span> <span m="3834300">sets</span> <span m="3834780">S</span> <span m="3835650">that</span> <span m="3835870">consists</span> <span m="3838550">of</span> <span m="3840630">the</span> <span m="3840740">first</span> <span m="3841660">m</span> <span m="3843370">selected</span> <span m="3844030">edges,</span> <span m="3852730">there</span> <span m="3853000">exists</span> <span m="3854320">a</span> <span m="3854340">minimum</span> <span m="3854690">spanning</span> <span m="3855130">tree</span> <span m="3860010">of</span> <span m="3860280">G</span> <span m="3861730">such</span> <span m="3862120">that</span> <span m="3866110">S</span> <span m="3866420">is</span> <span m="3866610">a</span> <span m="3866660">subset</span> <span m="3867130">of</span> <span m="3867590">E.</span> </p>
<p><span m="3868780">So</span> <span m="3868950">this</span> <span m="3869160">is</span> <span m="3869280">how</span> <span m="3869400">induction</span> <span m="3869900">hypothesis.</span> <span m="3871020">And</span> <span m="3871710">as</span> <span m="3871920">always,</span> <span m="3872420">we</span> <span m="3872510">start</span> <span m="3872810">with</span> <span m="3872950">the</span> <span m="3873030">base</span> <span m="3873390">case.</span> <span m="3874720">And</span> <span m="3875130">the</span> <span m="3875210">base</span> <span m="3875550">case</span> <span m="3875770">is</span> <span m="3875890">going</span> <span m="3876090">to</span> <span m="3876190">be</span> <span m="3876720">pretty</span> <span m="3876980">straightforward</span> <span m="3877830">in</span> <span m="3877940">this</span> <span m="3878110">particular</span> <span m="3878630">case.</span> </p>
<p></p>
<p><span m="3882750">We</span> <span m="3882900">have</span> <span m="3883080">to</span> <span m="3883120">do</span> <span m="3883260">a</span> <span m="3883290">little</span> <span m="3883480">bit</span> <span m="3883630">of</span> <span m="3883760">work</span> <span m="3883980">for</span> <span m="3884300">it.</span> <span m="3884690">So</span> <span m="3884910">let's</span> <span m="3885100">see</span> <span m="3885210">how</span> <span m="3885330">that</span> <span m="3885510">works.</span> </p>
<p><span m="3886660">So</span> <span m="3888110">the</span> <span m="3888210">base</span> <span m="3888520">case is</span> <span m="3888900">m</span> <span m="3889090">equals</span> <span m="3889470">0.</span> <span m="3890620">So</span> <span m="3890840">let's</span> <span m="3891290">reread</span> <span m="3891920">this</span> <span m="3892020">statement</span> <span m="3892570">where</span> <span m="3892780">m</span> <span m="3893050">is</span> <span m="3893230">equal</span> <span m="3893470">to</span> <span m="3893560">0.</span> <span m="3894880">Well,</span> <span m="3896130">if</span> <span m="3896440">we</span> <span m="3896600">have</span> <span m="3897300">zero</span> <span m="3897930">selected</span> <span m="3898470">edges,</span> <span m="3899710">then</span> <span m="3900060">S</span> <span m="3900230">is the</span> <span m="3900673">empty</span> <span m="3901116">set.</span> </p>
<p><span m="3903780">So</span> <span m="3903970">I</span> <span m="3904040">have</span> <span m="3904130">to</span> <span m="3904230">prove</span> <span m="3904540">that</span> <span m="3904640">for</span> <span m="3904810">all</span> <span m="3905120">G,</span> <span m="3905900">there</span> <span m="3906290">exists</span> <span m="3906840">a</span> <span m="3906990">minimum</span> <span m="3907240">spanning</span> <span m="3907700">tree</span> <span m="3908830">such</span> <span m="3909240">that</span> <span m="3909440">the</span> <span m="3909550">empty</span> <span m="3909860">set</span> <span m="3910430">is</span> <span m="3910740">a</span> <span m="3910810">subset</span> <span m="3911300">of</span> <span m="3911510">E.</span> <span m="3911800">Well,</span> <span m="3912085">the</span> <span m="3912370">empty</span> <span m="3912670">set</span> <span m="3912870">is</span> <span m="3913010">always</span> <span m="3913400">a</span> <span m="3913460">subset</span> <span m="3913910">of</span> <span m="3914120">E,</span> <span m="3914445">right?</span> <span m="3914770">So</span> <span m="3914900">that's</span> <span m="3915110">no</span> <span m="3915270">problem.</span> </p>
<p><span m="3918370">So</span> <span m="3918500">what</span> <span m="3918640">I</span> <span m="3918700">need</span> <span m="3918880">to</span> <span m="3918990">proof</span> <span m="3919340">is</span> <span m="3919420">that</span> <span m="3919620">for</span> <span m="3919790">all</span> <span m="3920090">G,</span> <span m="3920570">there</span> <span m="3920825">exists</span> <span m="3921080">a</span> <span m="3921450">minimum</span> <span m="3921770">spanning</span> <span m="3922310">tree</span> <span m="3923050">of</span> <span m="3923290">G.</span> <span m="3925780">And</span> <span m="3926350">this</span> <span m="3926710">is</span> <span m="3927030">where our</span> <span m="3927260">previous</span> <span m="3927610">theorem</span> <span m="3928110">comes</span> <span m="3928450">in,</span> <span m="3928610">because</span> <span m="3929080">we</span> <span m="3929180">showed</span> <span m="3929720">that</span> <span m="3929940">for</span> <span m="3930300">all</span> <span m="3930670">graphs</span> <span m="3931220">G,</span> <span m="3932010">there</span> <span m="3932190">exists</span> <span m="3932780">a</span> <span m="3932880">spanning</span> <span m="3933360">tree.</span> <span m="3933790">And</span> <span m="3933980">if</span> <span m="3934070">there</span> <span m="3934210">exists a</span> <span m="3934390">spanning</span> <span m="3934800">tree,</span> <span m="3934950">there</span> <span m="3935270">exists</span> <span m="3935810">a</span> <span m="3935920">minimum</span> <span m="3936250">weight</span> <span m="3936480">spanning</span> <span m="3936860">tree.</span> </p>
<p><span m="3938540">So</span> <span m="3938700">that's</span> <span m="3938960">the</span> <span m="3939030">base</span> <span m="3939400">case.</span> <span m="3939660">So</span> <span m="3939840">m</span> <span m="3940050">equals</span> <span m="3940380">0</span> <span m="3940690">implies</span> <span m="3942130">that</span> <span m="3942790">S</span> <span m="3943170">is</span> <span m="3943400">equal</span> <span m="3943680">to</span> <span m="3943820">the</span> <span m="3944060">empty</span> <span m="3944390">set.</span> <span m="3946380">So</span> <span m="3946470">that</span> <span m="3946650">means</span> <span m="3946940">that</span> <span m="3947120">S</span> <span m="3947330">is</span> <span m="3947500">definitely</span> <span m="3948530">a</span> <span m="3948610">subset</span> <span m="3949270">for</span> <span m="3949790">any</span> <span m="3953610">set</span> <span m="3953870">of</span> <span m="3954030">edges</span> <span m="3954500">of</span> <span m="3954760">a</span> <span m="3956910">minimum</span> <span m="3957180">weight</span> <span m="3957450">spanning</span> <span m="3957935">tree.</span> </p>
<p></p>
<p><span m="3962300">And</span> <span m="3962510">now</span> <span m="3962680">we're</span> <span m="3962850">going</span> <span m="3963060">to</span> <span m="3963210">use</span> <span m="3963540">our</span> <span m="3963780">theorem</span> <span m="3964380">to</span> <span m="3964510">show</span> <span m="3965130">that</span> <span m="3966150">there</span> <span m="3966580">also</span> <span m="3967030">exists</span> <span m="3967780">a</span> <span m="3967890">minimum</span> <span m="3968130">spanning</span> <span m="3968440">tree.</span> <span m="3968570">So</span> <span m="3968760">this</span> <span m="3969180">in</span> <span m="3969320">itself</span> <span m="3969660">is</span> <span m="3969770">not</span> <span m="3969910">yet</span> <span m="3970080">sufficient,</span> <span m="3970690">right?</span> <span m="3970950">You</span> <span m="3971110">see</span> <span m="3971370">that.</span> <span m="3973970">I</span> <span m="3974090">know</span> <span m="3974310">that</span> <span m="3976060">if</span> <span m="3976370">S</span> <span m="3976580">is</span> <span m="3976700">the</span> <span m="3976810">empty</span> <span m="3977040">set,</span> <span m="3977450">then</span> <span m="3977630">I</span> <span m="3977670">know</span> <span m="3977850">that this</span> <span m="3978190">always</span> <span m="3978580">holds.</span> <span m="3979120">But</span> <span m="3979180">the</span> <span m="3979260">statement</span> <span m="3979525">is</span> <span m="3979790">a</span> <span m="3979820">little</span> <span m="3979990">bit</span> <span m="3980370">bigger.</span> <span m="3981230">For</span> <span m="3981550">all</span> <span m="3981880">G,</span> <span m="3982740">I</span> <span m="3983170">still</span> <span m="3983390">need</span> <span m="3983520">to</span> <span m="3983610">prove</span> <span m="3983800">there</span> <span m="3984160">exists</span> <span m="3984320">a</span> <span m="3984390">minimum</span> <span m="3984690">spanning</span> <span m="3985060">tree.</span> <span m="3985280">And</span> <span m="3985360">that's</span> <span m="3985540">what our</span> <span m="3985890">previous</span> <span m="3986340">theorem</span> <span m="3986700">told</span> <span m="3987000">us.</span> <span m="3987950">So</span> <span m="3988200">we'll</span> <span m="3988280">now</span> <span m="3988350">write</span> <span m="3988530">that</span> <span m="3988760">part</span> <span m="3989000">out.</span> <span m="3989800">So</span> <span m="3989980">we</span> <span m="3990070">have</span> <span m="3990200">to</span> <span m="3990290">base</span> <span m="3990640">case.</span> </p>
<p></p>
<p><span m="3995130">Let's</span> <span m="3995630">see.</span> </p>
<p></p>
<p><span m="4002680">I</span> <span m="4003140">think</span> <span m="4003900">I</span> <span m="4004040">will</span> <span m="4004360">go</span> <span m="4006130">and</span> <span m="4006390">prove</span> <span m="4006480">this</span> <span m="4006740">over</span> <span m="4006890">here.</span> <span m="4009020">So</span> <span m="4009250">let's</span> <span m="4009810">look</span> <span m="4010060">at</span> <span m="4010180">the--</span> </p>
<p></p>
<p><span m="4013840">That's</span> <span m="4014010">the</span> <span m="4014160">inductive</span> <span m="4014610">step.</span> <span m="4015850">And</span> <span m="4016130">then</span> <span m="4016290">hopefully</span> <span m="4016770">we</span> <span m="4016910">can</span> <span m="4018270">play</span> <span m="4018690">a little</span> <span m="4018950">bit with</span> <span m="4019210">this</span> <span m="4019390">picture</span> <span m="4019750">up</span> <span m="4019900">here</span> <span m="4020620">to</span> <span m="4020970">get</span> <span m="4020990">some</span> <span m="4021453">insight.</span> </p>
<p></p>
<p><span m="4026550">So</span> <span m="4026800">how</span> <span m="4026910">do</span> <span m="4027000">we</span> <span m="4027110">do</span> <span m="4027280">this?</span> <span m="4029050">Well,</span> <span m="4029470">with</span> <span m="4029620">an</span> <span m="4029720">inductive</span> <span m="4030140">step,</span> <span m="4032310">we</span> <span m="4032420">start</span> <span m="4032850">as</span> <span m="4033080">usual.</span> </p>
<p></p>
<p><span m="4037880">We assume</span> <span m="4041140">that</span> <span m="4042730">P</span> <span m="4043000">of</span> <span m="4043270">m</span> <span m="4043350">holds.</span> </p>
<p><span m="4046390">Now</span> <span m="4046570">how</span> <span m="4046720">do</span> <span m="4046830">we</span> <span m="4046950">go</span> <span m="4047200">ahead?</span> <span m="4047880">So</span> <span m="4048560">I</span> <span m="4048670">want</span> <span m="4048860">to</span> <span m="4048960">prove</span> <span m="4049670">Pm</span> <span m="4050280">plus</span> <span m="4050570">1,</span> <span m="4051860">which</span> <span m="4052190">is</span> <span m="4052860">stated</span> <span m="4053280">over</span> <span m="4053580">here.</span> <span m="4054160">So</span> <span m="4056130">I</span> <span m="4056250">want</span> <span m="4056390">to</span> <span m="4056490">consider</span> <span m="4057200">the</span> <span m="4057320">set</span> <span m="4057680">S</span> <span m="4058040">that</span> <span m="4058150">consists</span> <span m="4058630">of</span> <span m="4058760">the</span> <span m="4058840">first</span> <span m="4059240">n</span> <span m="4059400">plus</span> <span m="4059660">1</span> <span m="4059910">selected</span> <span m="4060500">edges.</span> </p>
<p><span m="4060960">So</span> <span m="4062120">well,</span> <span m="4063120">essentially</span> <span m="4063770">what</span> <span m="4065290">I'm</span> <span m="4065410">interested</span> <span m="4065970">in</span> <span m="4066140">is</span> <span m="4066290">what</span> <span m="4066450">happens</span> <span m="4067050">in</span> <span m="4067210">the</span> <span m="4067290">n</span> <span m="4067480">plus</span> <span m="4067770">1</span> <span m="4068290">step.</span> <span m="4069940">So</span> <span m="4072900">let</span> <span m="4073340">E</span> <span m="4074560">denote</span> <span m="4077260">the</span> <span m="4077470">edge</span> <span m="4080030">that</span> <span m="4080320">I</span> <span m="4080610">added</span> <span m="4081250">in the</span> <span m="4081570">m</span> <span m="4082060">plus</span> <span m="4082350">1</span> <span m="4083002">step.</span> </p>
<p></p>
<p><span m="4090720">And</span> <span m="4090880">let</span> <span m="4091150">S</span> <span m="4092240">be</span> <span m="4094430">the</span> <span m="4094510">first</span> <span m="4095040">m</span> <span m="4095270">selected</span> <span m="4095770">edges.</span> <span m="4096319">And we</span> <span m="4096450">know</span> <span m="4096750">that</span> <span m="4096850">we can</span> <span m="4097020">apply</span> <span m="4097939">p</span> <span m="4098170">of</span> <span m="4098330">m</span> <span m="4098830">for S,</span> <span m="4099109">right?</span> <span m="4099500">Because</span> <span m="4099819">we</span> <span m="4100410">assume</span> <span m="4100740">this</span> <span m="4101100">is</span> <span m="4101220">our</span> <span m="4101500">inductive</span> <span m="4101890">step.</span> </p>
<p><span m="4102960">So</span> <span m="4103220">we</span> <span m="4103319">can</span> <span m="4103450">apply</span> <span m="4103770">something</span> <span m="4104120">for</span> <span m="4104310">S.</span> <span m="4105000">We</span> <span m="4105140">would</span> <span m="4105279">like</span> <span m="4105520">to</span> <span m="4105609">show</span> <span m="4105950">Pm</span> <span m="4106290">plus</span> <span m="4106550">1.</span> <span m="4107600">So</span> <span m="4107740">we</span> <span m="4107810">would</span> <span m="4107970">like</span> <span m="4108189">to</span> <span m="4108310">show</span> <span m="4108600">something</span> <span m="4108970">for</span> <span m="4109140">the</span> <span m="4109260">union</span> <span m="4109600">of</span> <span m="4109729">those</span> <span m="4110029">two.</span> <span m="4110710">So</span> <span m="4110880">let</span> <span m="4110990">me</span> <span m="4111100">repeat</span> <span m="4111425">that.</span> </p>
<p><span m="4113870">Let</span> <span m="4114120">S</span> <span m="4114550">denote</span> <span m="4116029">the</span> <span m="4116149">first</span> <span m="4119310">m</span> <span m="4119779">selected</span> <span m="4120340">edges.</span> </p>
<p></p>
<p><span m="4124330">So</span> <span m="4126470">by</span> <span m="4126800">our</span> <span m="4127100">induction</span> <span m="4127425">hypothesis,</span> <span m="4129800">we</span> <span m="4130050">can--</span> <span m="4132130">well,</span> <span m="4132609">let's</span> <span m="4132819">apply</span> <span m="4133229">it.</span> <span m="4134430">We</span> <span m="4134560">know</span> <span m="4135520">that</span> <span m="4135680">there</span> <span m="4135890">exists</span> <span m="4136779">a</span> <span m="4136960">minimum</span> <span m="4137319">spanning</span> <span m="4137810">tree</span> <span m="4138050">such</span> <span m="4138200">that</span> <span m="4138300">S</span> <span m="4138859">is</span> <span m="4139029">a</span> <span m="4139080">subset</span> <span m="4139819">of</span> <span m="4140260">the</span> <span m="4140520">edges.</span> </p>
<p><span m="4141700">So</span> <span m="4141859">let's</span> <span m="4142109">just</span> <span m="4142350">pick</span> <span m="4142560">one</span> <span m="4142779">such</span> <span m="4143229">minimum</span> <span m="4143500">spanning</span> <span m="4143950">tree.</span> <span m="4146050">So</span> <span m="4146359">let</span> <span m="4147729">T</span> <span m="4148020">star</span> <span m="4148450">be such</span> <span m="4149050">a</span> <span m="4149170">minimum</span> <span m="4149500">spanning</span> <span m="4149939">tree.</span> <span m="4151490">It</span> <span m="4151600">has</span> <span m="4152410">edges</span> <span m="4152960">E</span> <span m="4153170">star.</span> </p>
<p></p>
<p><span m="4163710">And</span> <span m="4163939">we</span> <span m="4164040">know</span> <span m="4164470">that</span> <span m="4164950">S</span> <span m="4165060">is</span> <span m="4165290">actually</span> <span m="4165870">a</span> <span m="4165950">subset</span> <span m="4166609">of</span> <span m="4166700">E</span> <span m="4167149">star.</span> </p>
<p></p>
<p><span m="4174890">Let's</span> <span m="4175710">look</span> <span m="4175979">at</span> <span m="4176100">the</span> <span m="4176189">very</span> <span m="4176550">first</span> <span m="4177160">case.</span> <span m="4177620">So</span> <span m="4177950">what</span> <span m="4178010">are the</span> <span m="4178120">kind</span> <span m="4178290">of</span> <span m="4178399">cases</span> <span m="4178870">that</span> <span m="4179010">we are</span> <span m="4179319">interested</span> <span m="4179579">in?</span> <span m="4179840">So</span> <span m="4180560">let's</span> <span m="4180630">think</span> <span m="4180939">again.</span> <span m="4181700">What</span> <span m="4181729">do</span> <span m="4181819">we</span> <span m="4181920">need</span> <span m="4182100">to</span> <span m="4182189">prove?</span> <span m="4182990">We</span> <span m="4183069">need</span> <span m="4183220">to</span> <span m="4183319">prove</span> <span m="4183670">Pn</span> <span m="4184160">plus</span> <span m="4184430">1.</span> </p>
<p><span m="4185420">So</span> <span m="4185600">we</span> <span m="4185700">need</span> <span m="4185870">to</span> <span m="4185970">prove</span> <span m="4186319">something</span> <span m="4186950">about</span> <span m="4187290">the</span> <span m="4187399">union</span> <span m="4187700">of</span> <span m="4187830">S</span> <span m="4188229">and</span> <span m="4188430">E.</span> <span m="4188670">We want</span> <span m="4188810">to</span> <span m="4188899">show</span> <span m="4189740">that</span> <span m="4189859">there</span> <span m="4190020">is</span> <span m="4190160">a</span> <span m="4190200">minimum</span> <span m="4190590">spanning</span> <span m="4191130">tree</span> <span m="4192310">such</span> <span m="4192729">that</span> <span m="4193410">the</span> <span m="4193640">edge</span> <span m="4194080">set</span> <span m="4194580">of</span> <span m="4195090">this</span> <span m="4195300">minimum</span> <span m="4195580">spanning</span> <span m="4195970">tree</span> <span m="4196220">contains</span> <span m="4196930">both</span> <span m="4197270">S</span> <span m="4198300">and</span> <span m="4198490">also</span> <span m="4198800">E.</span> </p>
<p><span m="4201190">So</span> <span m="4201550">what</span> <span m="4201750">kind</span> <span m="4201940">of</span> <span m="4202420">cases</span> <span m="4202890">can</span> <span m="4203100">we</span> <span m="4203580">handle</span> <span m="4204050">here?</span> <span m="4205100">Well,</span> <span m="4205360">what will</span> <span m="4205580">be</span> <span m="4205790">a</span> <span m="4206940">really</span> <span m="4207230">good</span> <span m="4207470">first</span> <span m="4207890">case?</span> <span m="4209590">The</span> <span m="4209690">first</span> <span m="4209990">case</span> <span m="4210350">could</span> <span m="4210580">be</span> <span m="4211300">that</span> <span m="4211600">E</span> <span m="4211950">is</span> <span m="4212230">actually</span> <span m="4212750">already</span> <span m="4213260">part</span> <span m="4213650">of</span> <span m="4213890">E</span> <span m="4214020">star.</span> <span m="4215790">Well,</span> <span m="4215980">if</span> <span m="4216050">that's</span> <span m="4216730">true,</span> <span m="4217100">then</span> <span m="4217290">of</span> <span m="4217460">course,</span> <span m="4218410">S</span> <span m="4218990">together</span> <span m="4219530">with</span> <span m="4219790">E</span> <span m="4220980">is</span> <span m="4221220">also</span> <span m="4222140">a</span> <span m="4222250">subset</span> <span m="4222780">of</span> <span m="4222940">E</span> <span m="4223170">star.</span> <span m="4224700">And</span> <span m="4225510">that</span> <span m="4225750">means</span> <span m="4228430">that we</span> <span m="4228600">are</span> <span m="4228750">done,</span> <span m="4229650">in</span> <span m="4229790">this</span> <span m="4229960">particular</span> <span m="4230430">case.</span> </p>
<p><span m="4232450">And</span> <span m="4232580">why is</span> <span m="4232920">this?</span> <span m="4233570">Because</span> <span m="4235190">this</span> <span m="4236110">is</span> <span m="4236330">an</span> <span m="4236410">example</span> <span m="4236960">of a</span> <span m="4237060">minimum</span> <span m="4237390">spanning</span> <span m="4237820">tree</span> <span m="4238050">that</span> <span m="4238220">contains</span> <span m="4238790">both</span> <span m="4239100">S</span> <span m="4239400">and</span> <span m="4239570">E.</span> <span m="4239800">So</span> <span m="4239980">it</span> <span m="4240070">contains</span> <span m="4240470">the</span> <span m="4240940">n plus</span> <span m="4241110">1</span> <span m="4241550">first</span> <span m="4242140">selected</span> <span m="4242690">edges.</span> <span m="4243430">And</span> <span m="4243580">so</span> <span m="4243720">this</span> <span m="4243890">is</span> <span m="4244020">Pn</span> <span m="4244340">plus</span> <span m="4244560">1.</span> </p>
<p><span m="4245830">Now</span> <span m="4245960">the</span> <span m="4246050">next</span> <span m="4246430">case is</span> <span m="4246880">the</span> <span m="4246940">difficult</span> <span m="4247400">part.</span> </p>
<p></p>
<p><span m="4252350">OK,</span> <span m="4252720">let's</span> <span m="4255760">wipe</span> <span m="4256150">out--</span> <span m="4258300">actually,</span> <span m="4258920">we</span> <span m="4259400">do</span> <span m="4259500">not</span> <span m="4259720">really</span> <span m="4260130">need</span> <span m="4260430">the</span> <span m="4260660">proof</span> <span m="4260800">of the</span> <span m="4260910">theorem</span> <span m="4261350">anymore.</span> <span m="4263150">So</span> <span m="4263410">let's</span> <span m="4263870">take</span> <span m="4265070">this</span> <span m="4265240">off</span> <span m="4265500">the</span> <span m="4265580">blackboard.</span> </p>
<p></p>
<p><span m="4272910">So</span> <span m="4273230">the</span> <span m="4273390">way</span> <span m="4273550">to</span> <span m="4273720">do</span> <span m="4273940">this</span> <span m="4274190">is</span> <span m="4274400">to</span> <span m="4277690">sort</span> <span m="4278040">of</span> <span m="4278130">see</span> <span m="4278910">how</span> <span m="4279190">we</span> <span m="4279390">can</span> <span m="4280590">use</span> <span m="4281650">the</span> <span m="4282740">tree,</span> <span m="4283210">T</span> <span m="4283470">star,</span> <span m="4284590">and</span> <span m="4284800">somehow</span> <span m="4285280">transform</span> <span m="4285930">it</span> <span m="4286040">into</span> <span m="4286330">another</span> <span m="4286680">tree</span> <span m="4287450">for</span> <span m="4287620">this</span> <span m="4287820">particular</span> <span m="4288380">case</span> <span m="4289680">where</span> <span m="4290000">we</span> <span m="4290315">assume</span> <span m="4291170">that</span> <span m="4291450">E</span> <span m="4291910">is</span> <span m="4292250">not</span> <span m="4292810">in</span> <span m="4293100">E</span> <span m="4293270">star.</span> </p>
<p></p>
<p><span m="4299950">So</span> <span m="4300160">let's</span> <span m="4300370">have</span> <span m="4300550">a</span> <span m="4300630">look</span> <span m="4300880">at</span> <span m="4300970">that</span> <span m="4301370">graph</span> <span m="4302700">up</span> <span m="4302830">there.</span> <span m="4304070">Let</span> <span m="4304180">me</span> <span m="4304360">rewrite</span> <span m="4304830">it</span> <span m="4304990">a</span> <span m="4305030">little</span> <span m="4305280">bit.</span> <span m="4305730">And</span> <span m="4306490">let's</span> <span m="4307940">look</span> <span m="4308230">at</span> <span m="4308390">what</span> <span m="4308540">happens</span> <span m="4309000">in</span> <span m="4309080">the</span> <span m="4309210">algorithm</span> <span m="4310020">after</span> <span m="4310380">two</span> <span m="4310600">steps.</span> </p>
<p><span m="4311200">So</span> <span m="4311490">I will</span> <span m="4312280">redraw</span> <span m="4312690">the</span> <span m="4312810">graph.</span> <span m="4314980">So</span> <span m="4315410">we</span> <span m="4315550">have,</span> <span m="4320170">say,</span> <span m="4323000">these</span> <span m="4323520">edges.</span> </p>
<p></p>
<p><span m="4333970">This</span> <span m="4334290">one--</span> <span m="4336595">let</span> <span m="4337056">me see.</span> <span m="4339370">I</span> <span m="4339500">want</span> <span m="4339680">to</span> <span m="4339800">use</span> <span m="4340020">some</span> <span m="4340160">different</span> <span m="4340570">colors</span> <span m="4342102">to</span> <span m="4342518">make</span> <span m="4342934">this</span> <span m="4343660">as</span> <span m="4343860">clear as</span> <span m="4344340">possible.</span> <span m="4346630">We</span> <span m="4346720">have</span> <span m="4347040">this</span> <span m="4347330">one.</span> <span m="4348623">We</span> <span m="4349056">have</span> <span m="4349490">this</span> <span m="4349780">one.</span> <span m="4350516">We</span> <span m="4350872">have</span> <span m="4351230">this</span> <span m="4351580">one.</span> <span m="4353170">And</span> <span m="4353550">we</span> <span m="4353760">also</span> <span m="4354080">have</span> <span m="4354280">this</span> <span m="4354490">one</span> <span m="4354650">actually.</span> <span m="4355140">Sorry.</span> </p>
<p><span m="4356440">All</span> <span m="4356590">right,</span> <span m="4356840">so</span> <span m="4358250">let's</span> <span m="4358560">define</span> <span m="4359030">the</span> <span m="4359110">different</span> <span m="4360200">edges.</span> <span m="4361740">The</span> <span m="4362100">straight</span> <span m="4362570">ones</span> <span m="4364500">from</span> <span m="4365050">the</span> <span m="4365140">set</span> <span m="4365480">S.</span> <span m="4366290">So</span> <span m="4366520">after</span> <span m="4366800">two</span> <span m="4367000">steps</span> <span m="4368020">over</span> <span m="4368290">there,</span> <span m="4368830">we have</span> <span m="4368980">selected</span> <span m="4369490">this</span> <span m="4369740">edge</span> <span m="4369990">first,</span> <span m="4370610">and</span> <span m="4370780">then</span> <span m="4370970">this</span> <span m="4371350">edge</span> <span m="4371730">was</span> <span m="4371920">the</span> <span m="4371990">second</span> <span m="4372310">edge</span> <span m="4372630">that we</span> <span m="4372930">selected</span> <span m="4373490">in</span> <span m="4373670">our</span> <span m="4374010">algorithm.</span> <span m="4375150">So</span> <span m="4375210">these</span> <span m="4375540">two</span> <span m="4375700">together</span> <span m="4377170">form</span> <span m="4377860">the</span> <span m="4378040">set</span> <span m="4378320">S.</span> </p>
<p><span m="4379860">So</span> <span m="4380090">what</span> <span m="4380280">is</span> <span m="4380470">T</span> <span m="4380810">star?</span> <span m="4382100">T</span> <span m="4382590">star</span> <span m="4383600">could</span> <span m="4383850">be,</span> <span m="4384170">for</span> <span m="4384220">example,</span> <span m="4384690">the</span> <span m="4384760">spanning</span> <span m="4385015">tree</span> <span m="4385270">that</span> <span m="4385700">contains</span> <span m="4386970">both</span> <span m="4388128">these</span> <span m="4388626">edges</span> <span m="4391620">plus</span> <span m="4393225">the</span> <span m="4393590">dotted</span> <span m="4393890">ones.</span> </p>
<p><span m="4396770">So</span> <span m="4396900">let's</span> <span m="4397110">have</span> <span m="4397270">a</span> <span m="4397350">look</span> <span m="4397530">here.</span> <span m="4398300">This</span> <span m="4398500">is</span> <span m="4398620">indeed</span> <span m="4398980">a</span> <span m="4399120">tree,</span> <span m="4399690">right?</span> <span m="4401390">And</span> <span m="4401940">if</span> <span m="4402300">we</span> <span m="4404250">look</span> <span m="4404480">into</span> <span m="4404850">this</span> <span m="4405740">picture</span> <span m="4406160">over</span> <span m="4406420">here,</span> <span m="4406580">you</span> <span m="4406750">can</span> <span m="4407260">see</span> <span m="4407550">that</span> <span m="4407720">the</span> <span m="4407830">weight</span> <span m="4408080">of</span> <span m="4408660">this</span> <span m="4408930">particular</span> <span m="4409410">tree</span> <span m="4409890">is</span> <span m="4410010">17.</span> <span m="4410710">It's</span> <span m="4410870">also</span> <span m="4411150">a</span> <span m="4411200">minimum</span> <span m="4411610">weight</span> <span m="4412350">spanning</span> <span m="4412830">tree.</span> </p>
<p><span m="4415890">What</span> <span m="4416100">is</span> <span m="4416240">G?</span> <span m="4417240">G</span> <span m="4417740">is the</span> <span m="4417900">complete</span> <span m="4418340">graph</span> <span m="4418910">and</span> <span m="4419100">contains</span> <span m="4419770">not</span> <span m="4420020">only</span> <span m="4421020">these</span> <span m="4421270">two</span> <span m="4421500">types</span> <span m="4421890">of</span> <span m="4422070">edges,</span> <span m="4423660">but</span> <span m="4423840">also</span> <span m="4424330">these</span> <span m="4425000">sort</span> <span m="4425200">of</span> <span m="4425340">more</span> <span m="4425560">orange</span> <span m="4425820">colored</span> <span m="4426282">ones.</span> </p>
<p></p>
<p><span m="4430910">So</span> <span m="4431110">this</span> <span m="4431360">is</span> <span m="4431530">so</span> <span m="4431770">the</span> <span m="4431870">situation</span> <span m="4432220">that</span> <span m="4432570">we</span> <span m="4432710">are in.</span> <span m="4434490">Now</span> <span m="4435150">in</span> <span m="4435360">our</span> <span m="4435600">algorithm,</span> <span m="4436530">we</span> <span m="4436690">want</span> <span m="4436890">to</span> <span m="4437050">choose</span> <span m="4438870">this</span> <span m="4439060">particular</span> <span m="4439720">third</span> <span m="4440270">edge.</span> <span m="4442350">So</span> <span m="4442470">this</span> <span m="4442750">is</span> <span m="4443700">going</span> <span m="4443960">to</span> <span m="4444090">be</span> <span m="4444340">our</span> <span m="4444730">e,</span> <span m="4447470">this</span> <span m="4447660">particular</span> <span m="4448090">edge.</span> <span m="4449140">And</span> <span m="4449320">as</span> <span m="4449390">you</span> <span m="4449870">can</span> <span m="4450050">see,</span> <span m="4450630">e</span> <span m="4450900">is</span> <span m="4451100">not</span> <span m="4451450">part</span> <span m="4452110">of</span> <span m="4452590">T star.</span> <span m="4453320">It's</span> <span m="4453540">not</span> <span m="4453850">one</span> <span m="4454190">of</span> <span m="4454370">those</span> <span m="4454690">two</span> <span m="4454920">kinds</span> <span m="4455210">of</span> <span m="4455520">edges.</span> <span m="4456190">It's</span> <span m="4456380">actually</span> <span m="4458010">one</span> <span m="4458140">of</span> <span m="4458230">the</span> <span m="4458330">orange</span> <span m="4458660">edges.</span> </p>
<p><span m="4460230">So</span> <span m="4460450">what</span> <span m="4460640">can</span> <span m="4460790">we</span> <span m="4460910">do</span> <span m="4461120">here?</span> <span m="4462400">Well,</span> <span m="4463160">the</span> <span m="4463320">idea</span> <span m="4463760">is</span> <span m="4464330">that</span> <span m="4464480">we</span> <span m="4465180">want</span> <span m="4465530">to</span> <span m="4465980">recreate</span> <span m="4466840">like</span> <span m="4467130">a</span> <span m="4467430">you</span> <span m="4469640">tree</span> <span m="4470430">in</span> <span m="4470610">which</span> <span m="4470890">we</span> <span m="4471040">somehow</span> <span m="4471450">exchange</span> <span m="4472310">e--</span> <span m="4474850">exchange</span> <span m="4475360">one</span> <span m="4475530">of</span> <span m="4475650">the</span> <span m="4475770">edges</span> <span m="4476440">in</span> <span m="4477160">T</span> <span m="4477410">star</span> <span m="4477990">with</span> <span m="4478250">e,</span> <span m="4478790">and</span> <span m="4478980">still</span> <span m="4479330">sort</span> <span m="4479620">of</span> <span m="4479720">maintaining</span> <span m="4480320">the</span> <span m="4480410">minimum</span> <span m="4480790">spanning</span> <span m="4481290">tree</span> <span m="4482020">property.</span> <span m="4482740">So</span> <span m="4482940">how</span> <span m="4483110">can</span> <span m="4483240">you</span> <span m="4483340">prove</span> <span m="4483630">this?</span> </p>
<p></p>
<p><span m="4492590">The</span> <span m="4492750">way</span> <span m="4492920">to</span> <span m="4493070">do</span> <span m="4493280">this</span> <span m="4493670">is</span> <span m="4493920">to</span> <span m="4495790">figure</span> <span m="4496500">out</span> <span m="4497190">what</span> <span m="4497430">the</span> <span m="4497590">algorithm</span> <span m="4498400">teaches</span> <span m="4498780">us</span> <span m="4499270">about</span> <span m="4499860">e,</span> <span m="4500610">and</span> <span m="4500830">also</span> <span m="4502080">what</span> <span m="4502630">the</span> <span m="4503100">tree</span> <span m="4503370">property</span> <span m="4503837">tells</span> <span m="4504304">us.</span> </p>
<p></p>
<p><span m="4508040">So</span> <span m="4508110">let's</span> <span m="4508320">combine</span> <span m="4508770">all</span> <span m="4509040">our</span> <span m="4509220">knowledge.</span> <span m="4510410">So</span> <span m="4510590">first</span> <span m="4510920">of</span> <span m="4511080">all,</span> <span m="4511220">we</span> <span m="4511340">know</span> <span m="4511530">that the</span> <span m="4511810">algorithm,</span> <span m="4513470">from</span> <span m="4513730">its</span> <span m="4513860">definition,</span> <span m="4514520">implies</span> <span m="4515600">that</span> <span m="4517570">as</span> <span m="4518040">together</span> <span m="4518470">with</span> <span m="4518870">e,</span> <span m="4519880">actually</span> <span m="4521960">has</span> <span m="4522130">no</span> <span m="4522400">cycle.</span> <span m="4524370">That's</span> <span m="4524530">how</span> <span m="4524720">we</span> <span m="4525700">select</span> <span m="4526160">the</span> <span m="4526330">n</span> <span m="4526500">plus</span> <span m="4526800">1</span> <span m="4527400">edge.</span> <span m="4528750">There's</span> <span m="4528980">no--</span> <span m="4531720">all</span> <span m="4531990">selected</span> <span m="4532460">edged</span> <span m="4532640">together</span> <span m="4532710">should</span> <span m="4533120">not</span> <span m="4533260">contain a</span> <span m="4533660">cycle.</span> <span m="4534020">That's the</span> <span m="4534260">definition</span> <span m="4534720">of the</span> <span m="4534780">algorithm.</span> </p>
<p><span m="4536040">We</span> <span m="4536160">know that</span> <span m="4536490">T</span> <span m="4536700">star</span> <span m="4537400">is</span> <span m="4537620">a</span> <span m="4537720">tree.</span> <span m="4539430">And</span> <span m="4540590">this</span> <span m="4540860">implies</span> <span m="4542210">that</span> <span m="4544600">if</span> <span m="4544800">you</span> <span m="4544940">look</span> <span m="4545170">at</span> <span m="4545260">the</span> <span m="4545350">graph</span> <span m="4545640">that</span> <span m="4545750">has</span> <span m="4546100">the</span> <span m="4547270">edges,</span> <span m="4547950">has</span> <span m="4548240">the</span> <span m="4548370">vertices</span> <span m="4548790">V</span> <span m="4548870">together</span> <span m="4549710">with</span> <span m="4550710">E</span> <span m="4550850">star</span> <span m="4552290">with</span> <span m="4552860">edge</span> <span m="4553450">e,</span> <span m="4554420">then</span> <span m="4554660">this</span> <span m="4554890">must</span> <span m="4555210">contain</span> <span m="4555330">a</span> <span m="4555790">cycle.</span> </p>
<p></p>
<p><span m="4559930">Now</span> <span m="4560280">this</span> <span m="4560630">property</span> <span m="4561140">I</span> <span m="4561180">will</span> <span m="4561340">not</span> <span m="4561770">prove</span> <span m="4562160">right</span> <span m="4562400">now,</span> <span m="4562660">but</span> <span m="4562770">it</span> <span m="4562900">is</span> <span m="4563070">pretty</span> <span m="4563330">straightforward.</span> <span m="4564000">It's</span> <span m="4564080">a</span> <span m="4564290">one line</span> <span m="4564630">proof</span> <span m="4564960">actually</span> <span m="4565330">in</span> <span m="4565420">the</span> <span m="4565500">book.</span> <span m="4565840">So you</span> <span m="4565980">can</span> <span m="4566140">look</span> <span m="4566300">it</span> <span m="4566400">up.</span> </p>
<p><span m="4567080">So</span> <span m="4567680">it</span> <span m="4567810">simply</span> <span m="4568130">states</span> <span m="4568610">that if</span> <span m="4568770">you</span> <span m="4568870">have</span> <span m="4569000">a</span> <span m="4569130">tree,</span> <span m="4569690">if</span> <span m="4569870">you</span> <span m="4570050">add</span> <span m="4570330">another</span> <span m="4570740">edge,</span> <span m="4571550">an</span> <span m="4572040">edge</span> <span m="4572180">that is</span> <span m="4572320">not</span> <span m="4573500">contained</span> <span m="4574000">in E</span> <span m="4574340">star,</span> <span m="4575500">then we</span> <span m="4575760">will</span> <span m="4576000">create</span> <span m="4576690">a</span> <span m="4576780">cycle.</span> </p>
<p><span m="4579110">So</span> <span m="4579310">now</span> <span m="4579510">we</span> <span m="4579630">can</span> <span m="4579780">combine</span> <span m="4580220">these</span> <span m="4580580">two</span> <span m="4580740">statements</span> <span m="4581310">together</span> <span m="4582780">and</span> <span m="4583100">conclude</span> <span m="4584320">that,</span> <span m="4586740">well,</span> <span m="4587160">if</span> <span m="4587720">S</span> <span m="4588080">with</span> <span m="4588360">e</span> <span m="4588970">has</span> <span m="4589250">no</span> <span m="4589430">cycle,</span> <span m="4590500">but</span> <span m="4590700">E</span> <span m="4590910">star</span> <span m="4591350">with</span> <span m="4591580">e</span> <span m="4591800">has</span> <span m="4592150">a</span> <span m="4592210">cycle,</span> <span m="4593740">and</span> <span m="4594050">S</span> <span m="4594430">is</span> <span m="4594630">contained</span> <span m="4595100">in</span> <span m="4595330">E</span> <span m="4595500">star,</span> <span m="4596940">well</span> <span m="4597050">there's</span> <span m="4597290">only</span> <span m="4597570">one</span> <span m="4597720">possibility.</span> <span m="4599180">And</span> <span m="4599310">that is</span> <span m="4599810">that</span> <span m="4602430">this</span> <span m="4602650">cycle</span> <span m="4605050">must</span> <span m="4605420">have</span> <span m="4605980">an</span> <span m="4606160">edge,</span> <span m="4610610">e</span> <span m="4610860">prime,</span> <span m="4612320">that</span> <span m="4612420">is</span> <span m="4612660">in</span> <span m="4613070">E</span> <span m="4613455">star</span> <span m="4614610">minus</span> <span m="4615310">S.</span> </p>
<p><span m="4616210">Because</span> <span m="4616610">if</span> <span m="4616720">it</span> <span m="4616820">would</span> <span m="4616970">not</span> <span m="4617160">have</span> <span m="4617310">such</span> <span m="4617590">an</span> <span m="4618690">edge,</span> <span m="4619630">then</span> <span m="4620220">all</span> <span m="4620700">the</span> <span m="4620910">edges</span> <span m="4621400">of</span> <span m="4621500">the</span> <span m="4621600">cycle</span> <span m="4622340">must</span> <span m="4622850">be</span> <span m="4623780">located</span> <span m="4624320">in</span> <span m="4624450">S--</span> <span m="4627670">in</span> <span m="4628230">S</span> <span m="4628440">together</span> <span m="4628790">with</span> <span m="4628940">E.</span> <span m="4629310">But</span> <span m="4629790">S</span> <span m="4630020">together</span> <span m="4630190">with</span> <span m="4630380">E</span> <span m="4630660">has</span> <span m="4630850">no</span> <span m="4630970">cycle.</span> <span m="4631500">So</span> <span m="4631510">that's</span> <span m="4631700">not</span> <span m="4631850">possible.</span> <span m="4632390">So</span> <span m="4632980">this</span> <span m="4633170">cycle</span> <span m="4633520">must</span> <span m="4633780">have</span> <span m="4633880">and</span> <span m="4634030">edge</span> <span m="4634290">e</span> <span m="4634460">prime</span> <span m="4634840">that</span> <span m="4634940">is</span> <span m="4635270">outside</span> <span m="4636580">S,</span> <span m="4636980">but</span> <span m="4637120">still</span> <span m="4637490">in E</span> <span m="4637840">star.</span> </p>
<p><span m="4638850">So</span> <span m="4639040">we</span> <span m="4639150">can</span> <span m="4639310">find</span> <span m="4639640">one</span> <span m="4639850">over</span> <span m="4640120">here.</span> <span m="4642000">e</span> <span m="4642200">prime</span> <span m="4642960">can</span> <span m="4643140">be</span> <span m="4643270">this</span> <span m="4643470">particular</span> <span m="4643920">edge.</span> <span m="4644800">So</span> <span m="4645050">e</span> <span m="4645260">prime</span> <span m="4645780">is</span> <span m="4646000">in E</span> <span m="4646300">star.</span> <span m="4648820">But</span> <span m="4648970">it is</span> <span m="4649290">not</span> <span m="4650340">already</span> <span m="4650670">selected</span> <span m="4651400">in</span> <span m="4651840">S.</span> </p>
<p><span m="4653980">So</span> <span m="4654130">now</span> <span m="4654780">we</span> <span m="4655000">are</span> <span m="4655090">going</span> <span m="4655380">to</span> <span m="4655510">do</span> <span m="4655660">a</span> <span m="4655760">trick.</span> <span m="4658830">The</span> <span m="4658940">idea</span> <span m="4659340">is</span> <span m="4659500">to</span> <span m="4660120">swap</span> <span m="4661500">e</span> <span m="4661790">and</span> <span m="4662010">e</span> <span m="4662230">prime.</span> <span m="4663560">So</span> <span m="4663720">that's</span> <span m="4663950">the</span> <span m="4664040">main</span> <span m="4664360">idea.</span> </p>
<p><span m="4665580">So</span> <span m="4665730">let's</span> <span m="4667230">first</span> <span m="4668220">write</span> <span m="4668530">down</span> <span m="4669380">what</span> <span m="4669540">we</span> <span m="4669700">know</span> <span m="4670180">about</span> <span m="4670620">the</span> <span m="4670740">weight</span> <span m="4671230">of</span> <span m="4671450">e</span> <span m="4671620">prime</span> <span m="4671990">and</span> <span m="4672150">e.</span> <span m="4673770">So</span> <span m="4674070">since</span> <span m="4674310">the</span> <span m="4674510">algorithm,</span> <span m="4678850">well,</span> <span m="4679100">could</span> <span m="4679300">have</span> <span m="4679460">selected</span> <span m="4683750">e</span> <span m="4684370">or</span> <span m="4684730">e</span> <span m="4684920">prime--</span> <span m="4685320">it</span> <span m="4685440">did select</span> <span m="4686020">e,</span> <span m="4686250">right?</span> <span m="4687300">But</span> <span m="4687440">it</span> <span m="4687560">could have</span> <span m="4687850">also</span> <span m="4688060">selected</span> <span m="4688490">e</span> <span m="4688740">prime,</span> <span m="4689670">because e</span> <span m="4690080">prime</span> <span m="4692700">does</span> <span m="4692990">not</span> <span m="4693280">create</span> <span m="4693700">a</span> <span m="4693770">cycle</span> <span m="4694370">with</span> <span m="4694620">S.</span> <span m="4695150">That's</span> <span m="4695350">what</span> <span m="4695580">we</span> <span m="4696980">have</span> <span m="4697120">seen</span> <span m="4697430">here.</span> <span m="4698760">So</span> <span m="4700630">this</span> <span m="4700890">really</span> <span m="4701210">implies</span> <span m="4701810">therefore</span> <span m="4702890">that</span> <span m="4703150">the</span> <span m="4703310">weight</span> <span m="4706120">of</span> <span m="4706470">e</span> <span m="4707920">is</span> <span m="4708040">at</span> <span m="4708230">most</span> <span m="4710300">the</span> <span m="4710450">weight</span> <span m="4712980">of</span> <span m="4713700">e</span> <span m="4713890">prime.</span> </p>
<p><span m="4715520">So</span> <span m="4715830">why</span> <span m="4715940">is</span> <span m="4716090">this?</span> <span m="4716210">Because</span> <span m="4716510">the</span> <span m="4716620">algorithm</span> <span m="4717090">always</span> <span m="4717580">chooses</span> <span m="4717800">the</span> <span m="4717900">minimum</span> <span m="4718370">weight.</span> <span m="4719540">And</span> <span m="4720270">it</span> <span m="4720440">can</span> <span m="4720600">choose</span> <span m="4720880">between</span> <span m="4721230">e or</span> <span m="4721630">e</span> <span m="4721760">prime,</span> <span m="4721960">but</span> <span m="4722170">it</span> <span m="4722370">chose</span> <span m="4723070">e.</span> <span m="4723515">So the</span> <span m="4723960">weight</span> <span m="4724220">of</span> <span m="4724320">e</span> <span m="4724530">must</span> <span m="4724760">be</span> <span m="4724870">less</span> <span m="4725190">than</span> <span m="4725360">the</span> <span m="4725470">weight</span> <span m="4725800">of</span> <span m="4725960">e</span> <span m="4726120">prime.</span> <span m="4727320">So</span> <span m="4727500">that's</span> <span m="4727720">what</span> <span m="4727870">we</span> <span m="4728000">know</span> <span m="4728190">also.</span> </p>
<p><span m="4729620">So</span> <span m="4729720">now</span> <span m="4729980">let's</span> <span m="4730810">swap</span> <span m="4732160">e</span> <span m="4732460">and</span> <span m="4732620">e</span> <span m="4732800">prime</span> <span m="4733110">in T.</span> <span m="4734030">And</span> <span m="4734190">then we're</span> <span m="4734420">almost</span> <span m="4734860">done</span> <span m="4735050">with</span> <span m="4735190">this</span> <span m="4735380">proof,</span> <span m="4735900">because</span> <span m="4739510">that</span> <span m="4739810">will</span> <span m="4739960">be</span> <span m="4740530">the</span> <span m="4740690">tree</span> <span m="4741190">of</span> <span m="4741450">which we will</span> <span m="4741710">prove</span> <span m="4742020">that</span> <span m="4742100">it is</span> <span m="4742490">a</span> <span m="4742964">minimum</span> <span m="4743438">spanning</span> <span m="4743912">tree.</span> </p>
<p></p>
<p><span m="4750870">So</span> <span m="4752460">the</span> <span m="4752630">key</span> <span m="4752900">idea</span> <span m="4753200">is</span> <span m="4753500">to</span> <span m="4753620">swap</span> <span m="4755560">e</span> <span m="4756690">and</span> <span m="4757300">e</span> <span m="4757590">prime</span> <span m="4759450">in</span> <span m="4759640">T.</span> <span m="4761220">How</span> <span m="4761360">do</span> <span m="4761470">we</span> <span m="4761580">do</span> <span m="4761770">it?</span> <span m="4762510">Well,</span> <span m="4762840">let</span> <span m="4764080">T</span> <span m="4764590">double</span> <span m="4764910">star</span> <span m="4765660">be</span> <span m="4766130">equal</span> <span m="4766560">to</span> <span m="4766900">V</span> <span m="4767390">with</span> <span m="4767910">the</span> <span m="4768230">edge</span> <span m="4768505">set</span> <span m="4770190">E</span> <span m="4770470">double</span> <span m="4770740">star</span> <span m="4771820">where</span> <span m="4772080">E</span> <span m="4773350">double</span> <span m="4773690">star</span> <span m="4774300">is</span> <span m="4774550">really</span> <span m="4774870">equal</span> <span m="4775380">to,</span> <span m="4776156">well,</span> <span m="4776950">the</span> <span m="4777100">original</span> <span m="4777840">set</span> <span m="4778320">E</span> <span m="4778530">star.</span> <span m="4779260">But</span> <span m="4779460">we</span> <span m="4779820">take</span> <span m="4780140">out of</span> <span m="4780410">this</span> <span m="4781410">e</span> <span m="4781600">prime.</span> <span m="4783000">So</span> <span m="4783250">this</span> <span m="4783520">one</span> <span m="4784190">is</span> <span m="4784380">taken</span> <span m="4784830">out</span> <span m="4785260">of</span> <span m="4785420">the</span> <span m="4785550">minimum</span> <span m="4785900">spanning</span> <span m="4786250">tree.</span> <span m="4786330">And we</span> <span m="4786500">add</span> <span m="4786830">e in</span> <span m="4788152">here.</span> </p>
<p></p>
<p><span m="4792310">Now</span> <span m="4792440">we</span> <span m="4792520">want</span> <span m="4792680">to</span> <span m="4792800">prove</span> <span m="4793270">that</span> <span m="4793470">this</span> <span m="4793650">particular</span> <span m="4794200">T</span> <span m="4794440">double</span> <span m="4794700">star</span> <span m="4795410">is</span> <span m="4795630">a</span> <span m="4795680">minimum</span> <span m="4796110">spanning</span> <span m="4796640">tree</span> <span m="4796940">that</span> <span m="4797100">fits</span> <span m="4797800">our</span> <span m="4797990">lemma.</span> <span m="4799810">So</span> <span m="4801540">what</span> <span m="4801660">do</span> <span m="4801750">we</span> <span m="4801880">know?</span> <span m="4803180">We</span> <span m="4803330">know that</span> <span m="4803690">T</span> <span m="4803900">double</span> <span m="4804200">star</span> <span m="4804550">is</span> <span m="4804900">acyclic.</span> </p>
<p></p>
<p><span m="4810500">And</span> <span m="4810650">why</span> <span m="4810850">is</span> <span m="4810990">this?</span> <span m="4812030">Well,</span> <span m="4813292">we</span> <span m="4813750">actually</span> <span m="4817240">removed</span> <span m="4817990">e</span> <span m="4818310">prime</span> <span m="4819640">from</span> <span m="4822740">the</span> <span m="4823060">only</span> <span m="4823460">cycle--</span> <span m="4824890">so</span> <span m="4825500">look</span> <span m="4825710">up</span> <span m="4825910">here</span> <span m="4826140">also--</span> <span m="4827010">the</span> <span m="4827160">only</span> <span m="4827460">cycle</span> <span m="4829070">in</span> <span m="4830650">E</span> <span m="4832230">star</span> <span m="4833000">together</span> <span m="4833610">with</span> <span m="4834000">e.</span> <span m="4835720">So</span> <span m="4835990">it's</span> <span m="4837170">acyclic.</span> </p>
<p><span m="4839110">T</span> <span m="4839500">double</span> <span m="4839760">star</span> <span m="4840260">is</span> <span m="4840470">also</span> <span m="4840780">connected.</span> <span m="4843660">And</span> <span m="4843840">why</span> <span m="4844020">is</span> <span m="4844150">that?</span> <span m="4845620">Well,</span> <span m="4847840">since</span> <span m="4848330">e</span> <span m="4848780">prime</span> <span m="4849340">is</span> <span m="4849530">removed--</span> <span m="4850260">but</span> <span m="4850660">it</span> <span m="4850780">was</span> <span m="4850980">on</span> <span m="4851100">a</span> <span m="4851150">cycle.</span> <span m="4852440">So</span> <span m="4852745">it</span> <span m="4853050">remains</span> <span m="4853519">connected.</span> </p>
<p></p>
<p><span m="4858680">And</span> <span m="4858990">finally,</span> <span m="4860070">we</span> <span m="4860210">also</span> <span m="4860520">know</span> <span m="4860900">that</span> <span m="4861070">T</span> <span m="4862200">double</span> <span m="4862633">star</span> <span m="4863910">actually</span> <span m="4864300">contains</span> <span m="4866180">all</span> <span m="4868320">the</span> <span m="4868440">vertices</span> <span m="4871960">in</span> <span m="4872280">G.</span> <span m="4873400">So</span> <span m="4873830">all</span> <span m="4874080">these</span> <span m="4874270">three</span> <span m="4874510">together</span> <span m="4875530">with</span> <span m="4875580">prove</span> <span m="4876900">that</span> <span m="4877080">T</span> <span m="4877150">double</span> <span m="4877500">star</span> <span m="4878300">is</span> <span m="4878670">a</span> <span m="4879550">spanning</span> <span m="4880140">tree</span> <span m="4881950">of</span> <span m="4882140">G,</span> <span m="4882830">because</span> <span m="4883150">it's</span> <span m="4883330">acyclic,</span> <span m="4884070">connected,</span> <span m="4885020">contains</span> <span m="4885430">all</span> <span m="4885570">the</span> <span m="4885650">vertices.</span> </p>
<p><span m="4886700">Is it</span> <span m="4886790">a</span> <span m="4887120">minimum</span> <span m="4887380">weight</span> <span m="4887610">spanning</span> <span m="4888000">tree?</span> <span m="4889780">Now</span> <span m="4890010">we</span> <span m="4890110">are</span> <span m="4890420">really</span> <span m="4891050">almost</span> <span m="4891520">done,</span> <span m="4892500">because--</span> <span m="4894020">let</span> <span m="4894200">me</span> <span m="4894320">write</span> <span m="4894530">it</span> <span m="4894670">out</span> <span m="4894840">here.</span> <span m="4895980">We</span> <span m="4896110">know</span> <span m="4896390">that</span> <span m="4897000">the</span> <span m="4897880">weight</span> <span m="4900480">of</span> <span m="4900890">T</span> <span m="4901200">double</span> <span m="4901470">star</span> <span m="4902980">is</span> <span m="4903570">at</span> <span m="4903640">most</span> <span m="4904310">the</span> <span m="4904720">weight</span> <span m="4906022">of</span> <span m="4906490">T</span> <span m="4906870">star.</span> </p>
<p><span m="4908270">Why</span> <span m="4908370">is</span> <span m="4908500">this?</span> <span m="4909210">Well,</span> <span m="4909580">we</span> <span m="4909990">showed</span> <span m="4910470">just</span> <span m="4910770">now</span> <span m="4911680">that</span> <span m="4911920">the</span> <span m="4912040">weight</span> <span m="4912350">of</span> <span m="4912510">e</span> <span m="4913570">is</span> <span m="4913870">at</span> <span m="4913990">most</span> <span m="4914290">the</span> <span m="4914340">weight</span> <span m="4914490">of</span> <span m="4914590">e</span> <span m="4914830">prime.</span> <span m="4915350">So</span> <span m="4915650">we</span> <span m="4915770">have</span> <span m="4916110">exchanged</span> <span m="4916850">e</span> <span m="4917170">and</span> <span m="4917340">e</span> <span m="4917540">prime.</span> <span m="4918360">So</span> <span m="4918430">that</span> <span m="4918640">means</span> <span m="4918910">that</span> <span m="4919500">the</span> <span m="4919610">weight</span> <span m="4919760">of</span> <span m="4919940">T</span> <span m="4920140">double</span> <span m="4920400">star</span> <span m="4920750">is</span> <span m="4921030">at</span> <span m="4921200">most</span> <span m="4921830">that</span> <span m="4922050">of</span> <span m="4922240">the</span> <span m="4922350">weight</span> <span m="4922530">of</span> <span m="4922650">T</span> <span m="4922860">star.</span> </p>
<p><span m="4924110">We</span> <span m="4924240">also</span> <span m="4924640">know</span> <span m="4925940">that</span> <span m="4926250">T</span> <span m="4928380">star</span> <span m="4930050">is</span> <span m="4930280">a</span> <span m="4930380">minimum</span> <span m="4930880">spanning</span> <span m="4931350">tree.</span> <span m="4932750">So</span> <span m="4932990">if</span> <span m="4933190">you</span> <span m="4933330">combine</span> <span m="4934760">these</span> <span m="4935060">two</span> <span m="4935480">together,</span> <span m="4937600">then</span> <span m="4937930">we</span> <span m="4938250">know that the</span> <span m="4938580">weight</span> <span m="4938760">of</span> <span m="4939030">T</span> <span m="4939130">double</span> <span m="4939330">star</span> <span m="4939580">cannot</span> <span m="4939850">get</span> <span m="4940010">less</span> <span m="4940290">than</span> <span m="4940390">the</span> <span m="4940470">minimum</span> <span m="4940850">weight</span> <span m="4941230">possibility.</span> <span m="4942010">So</span> <span m="4943360">it</span> <span m="4943670">also</span> <span m="4943940">has</span> <span m="4944080">a</span> <span m="4944120">minimum</span> <span m="4944400">weight.</span> <span m="4945080">So</span> <span m="4945290">T</span> <span m="4945560">double</span> <span m="4945880">star</span> <span m="4947170">is</span> <span m="4947750">a</span> <span m="4948660">minimum</span> <span m="4949040">spanning</span> <span m="4949510">tree.</span> </p>
<p><span m="4950940">So</span> <span m="4951100">now</span> <span m="4951350">we're</span> <span m="4951470">done</span> <span m="4951720">because</span> <span m="4952010">in</span> <span m="4952120">this</span> <span m="4952420">case,</span> <span m="4953250">we</span> <span m="4953380">have</span> <span m="4953570">constructed</span> <span m="4954170">a</span> <span m="4954330">T</span> <span m="4956310">double</span> <span m="4956650">star</span> <span m="4957260">such</span> <span m="4957720">that</span> <span m="4958360">S</span> <span m="4958830">together</span> <span m="4959980">with</span> <span m="4960480">E</span> <span m="4961310">is</span> <span m="4961530">a</span> <span m="4961590">subset</span> <span m="4962740">of</span> <span m="4962980">E</span> <span m="4963360">double</span> <span m="4963610">star.</span> <span m="4964720">And</span> <span m="4964910">that</span> <span m="4965160">means</span> <span m="4965530">that</span> <span m="4965660">we</span> <span m="4965790">have</span> <span m="4966030">shown</span> <span m="4967460">that</span> <span m="4968270">our</span> <span m="4970160">induction</span> <span m="4970630">step</span> <span m="4971440">is</span> <span m="4971650">true.</span> <span m="4972110">We</span> <span m="4972270">have</span> <span m="4972360">shown</span> <span m="4972600">that Pn</span> <span m="4973080">plus</span> <span m="4973290">1</span> <span m="4973490">holds.</span> </p>
<p><span m="4974800">So</span> <span m="4975010">now</span> <span m="4975280">we</span> <span m="4975390">finally</span> <span m="4975705">have</span> <span m="4976020">figured</span> <span m="4976350">out</span> <span m="4976520">that the</span> <span m="4976610">lemma</span> <span m="4976820">is</span> <span m="4977140">true.</span> <span m="4978040">And</span> <span m="4978400">now</span> <span m="4978540">we</span> <span m="4979680">showed</span> <span m="4980000">already</span> <span m="4980480">how</span> <span m="4980650">to</span> <span m="4980800">use</span> <span m="4981030">the</span> <span m="4981100">lemma</span> <span m="4981560">in our</span> <span m="4981810">proof</span> <span m="4985080">for</span> <span m="4985270">this</span> <span m="4985430">particular</span> <span m="4985780">theorem.</span> </p>
<p><span m="4987570">So</span> <span m="4987700">this</span> <span m="4987970">is</span> <span m="4990130">the</span> <span m="4990240">end</span> <span m="4990380">of</span> <span m="4990460">this</span> <span m="4990690">lecture.</span> <span m="4991030">And</span> <span m="4991640">tomorrow,</span> <span m="4992410">you will</span> <span m="4992590">actually</span> <span m="4992860">go</span> <span m="4993080">again</span> <span m="4993870">over</span> <span m="4994140">this</span> <span m="4994350">proof</span> <span m="4994620">because</span> <span m="4994950">it's</span> <span m="4995090">rather</span> <span m="4995360">complex.</span> <span m="4996830">And</span> <span m="4997100">then</span> <span m="4998070">hopefully</span> <span m="4998450">you</span> <span m="4998610">really</span> <span m="4998980">get</span> <span m="4999250">into</span> <span m="4999540">all</span> <span m="4999740">the</span> <span m="4999970">different proof</span> <span m="5000310">techniques.</span> </p>
<p><span m="5001290">OK.</span> <span m="5001610">Thank</span> <span m="5001960">you.</span> </p>
<p></p>
</div>
        <div id="vid_related" itemprop="description" class="tabContent hide">
<h2 class="subhead">Free Downloads</h2>
<h3 class="subsubhead">Video</h3>
<ul>
<li>iTunes U (<a href="http://itunes.apple.com/us/itunes-u/lecture-8-graph-theory-ii/id503873536?i=110644969">MP4 - 182MB</a>)</li>
<li>Internet Archive (<a href="http://www.archive.org/download/MIT6.042JF10/MIT6_042JF10_lec08_300k.mp4">MP4 - 182MB</a>)</li>
</ul>
<br>
<h3 class="subsubhead">Caption</h3>
<ul>
<li>English-US (<a href="../../../contents/video-lectures/lecture-8-graph-theory-ii-minimum-spanning-trees/GJpt_3ie4WU.srt">SRT</a>)</li>
</ul>
</div>
    
   </div>  




      					 
        <div class="" id="parent-fieldname-bottom_html_area">
            
            
        </div>
    
               </main><!--Course_inner_media tag close -->
           		</div>
<!--Course_wrapper tag close -->
            </div>
<!--left tag close -->
            <aside id="right">
                <!--Begin Right Portion -->
                    <div>
    



</div>

                	<div>
    



</div>


        <div class="" id="parent-fieldname-rsi_top_html_area">
            
            
        </div>
    

<!-- RSI google ad space-->



<div id="google_ads">    
    <script async="async" src="https://www.googletagservices.com/tag/js/gpt.js"></script>
    <script type="text/javascript">var googletag = googletag || {}; googletag.cmd = googletag.cmd || [];</script>
    <script type="text/javascript">
googletag.cmd.push(function() {googletag.defineSlot('/1064917/VIDEO_INDIVIDUAL_SLOT_A_DL', [[300, 250], [300, 300], [180, 200], [180, 150], [160, 600]], 'VIDEO_INDIVIDUAL_SLOT_A_DL').addService(googletag.pubads());googletag.defineSlot('/1064917/VIDEO_INDIVIDUAL_SLOT_B_DL', [[300, 250], [300, 300], [180, 200], [180, 150], [160, 600]], 'VIDEO_INDIVIDUAL_SLOT_B_DL').addService(googletag.pubads());googletag.defineSlot('/1064917/VIDEO_INDIVIDUAL_SLOT_C_DL', [[300, 250], [300, 300], [180, 200], [180, 150], [160, 600]], 'VIDEO_INDIVIDUAL_SLOT_C_DL').addService(googletag.pubads());
googletag.pubads().enableSingleRequest();
 googletag.enableServices();
});</script>
    <script language="javascript" type="text/javascript">
googletag.cmd.push(function() {googletag.pubads().set("TYPE","HOUSE");googletag.pubads().set("DEPARTMENT","6");googletag.pubads().set("CRS_BEG2","04");googletag.pubads().set("CRS_END","2J");googletag.pubads().set("SESSION","F");googletag.pubads().set("YEAR","10");})
</script>
    
    <div id="VIDEO_INDIVIDUAL_SLOT_A_DL">
    	<script>googletag.cmd.push(function() { googletag.display('VIDEO_INDIVIDUAL_SLOT_A_DL'); });</script>
    </div>
    <div id="VIDEO_INDIVIDUAL_SLOT_B_DL">
    	<script>googletag.cmd.push(function() { googletag.display('VIDEO_INDIVIDUAL_SLOT_B_DL'); });</script>
    </div>
    <div id="VIDEO_INDIVIDUAL_SLOT_C_DL">
    	<script>googletag.cmd.push(function() { googletag.display('VIDEO_INDIVIDUAL_SLOT_C_DL'); });</script>
    </div>
</div>

<!-- End RSI ads--> 


<div>
    



</div>

            </aside><!--Right div close -->
            <div class="clear"></div>
        </div>
<!--grid tag close -->
      </div>
		
		<footer id="bottom">
			<div id="grid">
				
<div id="portletwrapper-6f63772e626f74746f6d706f72746c65746d616e616765720a636f6e746578740a2f506c6f6e650a736974652d666f6f746572" class="portletWrapper kssattr-portlethash-6f63772e626f74746f6d706f72746c65746d616e616765720a636f6e746578740a2f506c6f6e650a736974652d666f6f746572">
<div class="portletStaticText portlet-static-site-footer"><div id="footer">
<nav aria-label="Footer">     <nav id="foot-c1" class="grid_2 alpha" aria-labelledby="f-find-courses">       <span class="footer" id="f-find-courses" aria-hidden="true">Find Courses</span>
<ul class="foot-bullet" role="presentation">
    <li><a href="https://ocw.mit.edu/courses/find-by-topic/">Find by Topic</a></li>
    <li><a href="https://ocw.mit.edu/courses/find-by-number/">Find by Course Number</a></li>
    <li><a href="https://ocw.mit.edu/courses/find-by-department/">Find by Department</a></li>
    <li><a href="https://ocw.mit.edu/courses/new-courses/">New Courses</a></li>
    <li><a href="https://ocw.mit.edu/courses/most-visited-courses/">Most Visited Courses</a></li>
    <li><a href="https://ocw.mit.edu/courses/ocw-scholar/">OCW Scholar Courses</a></li>
    <li><a href="https://ocw.mit.edu/courses/audio-video-courses/">Audio/Video Courses</a></li>
    <li><a href="https://ocw.mit.edu/courses/online-textbooks/">Online Textbooks</a></li>
    <li><a href="https://ocw.mit.edu/courses/instructor-insights/">Instructor Insights</a></li>
    <li><a href="https://ocw.mit.edu/resources/">Supplemental Resources</a></li>
    <li><a href="https://ocw.mit.edu/courses/mitx-related-courseware/">MITx &amp; Related OCW Courses</a></li>
    <li><a href="https://ocw.mit.edu/courses/mit-open-learning-library/">MIT Open Learning Library</a></li>
    <li><a href="https://ocw.mit.edu/courses/translated-courses/">Translated Courses</a></li>
</ul>
</nav>
<div id="foot-c2" class="grid_2"><nav aria-labelledby="f-for-educators">         <span id="f-for-educators" class="footer" aria-hidden="true">For Educators</span>
<ul class="foot-bullet" role="presentation">
    <li><a href="https://chalk-radio.simplecast.com/">Chalk Radio Podcast</a></li>
    <li><a href="https://ocw.mit.edu/educator/">OCW Educator Portal </a></li>
    <li><a href="https://ocw.mit.edu/courses/instructor-insights/">Instructor Insights by Department</a></li>
    <li><a href="https://openlearning.mit.edu/campus/digital-innovations/" aria-label="External Link: Residential Digital Innovations">Residential Digital Innovations </a></li>
    <li><a href="https://ocw.mit.edu/high-school/">OCW Highlights for High School</a></li>
    <li><a href="https://ocw.mit.edu/educator/additional-resources/">Additional Resources</a></li>
</ul>
</nav></div>
<nav class="grid_2" id="foot-c3" aria-labelledby="f-donate">       <span id="f-donate" class="footer" aria-hidden="true">Give Now</span>
<ul class="foot-bullet" role="presentation">
    <li><a href="https://ocw.mit.edu/give/">Make a Donation</a></li>
    <li><a href="https://ocw.mit.edu/give/why-give/">Why Give?</a></li>
    <li><a href="https://ocw.mit.edu/give/our-supporters/">Our Supporters</a></li>
    <li><a href="https://ocw.mit.edu/give/other-ways-to-contribute/">Other Ways to Contribute</a></li>
    <li><a href="https://ocw.mit.edu/support/">Become a Corporate Sponsor</a></li>
</ul>
</nav>
<div class="grid_2" id="foot-c4">
<nav aria-labelledby="f-about">         <span id="f-about" class="footer" aria-hidden="true">About</span>
<ul class="foot-bullet" role="presentation">
    <li><a href="https://ocw.mit.edu/about/">About OpenCourseWare</a></li>
    <li><a href="https://ocw.mit.edu/about/site-statistics/">Site Statistics</a></li>
    <li><a href="https://ocw.mit.edu/about/ocw-stories/">OCW Stories</a></li>
    <li><a href="https://ocw.mit.edu/about/newsletter/">Newsletter</a></li>
    <li><a href="https://www.ocw-openmatters.org/">Open Matters Blog</a></li>
</ul>
</nav><!--about-->       <nav aria-labelledby="f-tools">         <span id="f-tools" class="footer" aria-hidden="true">Tools</span>
<ul class="foot-bullet" role="presentation">
    <li><a href="https://ocw.mit.edu/help/">Help &amp; FAQs</a></li>
    <li><a href="https://ocw.mit.edu/about/contactus">Contact Us</a></li>
    <li><a href="https://accessibility.mit.edu/" target="_blank">Accessibility</a></li>
    <li><a href="https://ocw.mit.edu/help/site-map/">Site Map</a></li>
    <li><a href="../../../common/terms/index.htm">Privacy &amp; Terms of Use</a></li>
    <li><a href="https://ocw.mit.edu/help/rss/">RSS Feeds</a></li>
</ul>
</nav><!--tools-->
</div>
</nav> <aside style="min-height: 289px;" aria-labelledby="f-our-corporate-supporters" class="grid_4 omega" id="foot-c5">           <span aria-hidden="true" class="footer" id="f-our-corporate-supporters">Our Corporate Supporters</span>           <!-- HOME_CORP_LOGO_1 -->
<div id="div-gpt-ad-1388181177156-0" class="sponsors_google_ads_even"><script type="text/javascript">
              googletag.cmd.push(function() { googletag.display('div-gpt-ad-1388181177156-0'); });
            </script></div>
<!-- HOME_CORP_LOGO_2 -->
<div id="div-gpt-ad-1388181177156-1" class="sponsors_google_ads_odd"><script type="text/javascript">
              googletag.cmd.push(function() { googletag.display('div-gpt-ad-1388181177156-1'); });
            </script></div>
<!-- HOME_CORP_LOGO_3 -->
<div id="div-gpt-ad-1388181177156-2" class="sponsors_google_ads_even"><script type="text/javascript">
              googletag.cmd.push(function() { googletag.display('div-gpt-ad-1388181177156-2'); });
            </script></div>
<!-- HOME_CORP_LOGO_4 -->
<div id="div-gpt-ad-1388181177156-3" class="sponsors_google_ads_odd"><script type="text/javascript">
              googletag.cmd.push(function() { googletag.display('div-gpt-ad-1388181177156-3'); });
            </script></div>
<!-- HOME_CORP_LOGO_5 -->
<div id="div-gpt-ad-1388181177156-4" class="sponsors_google_ads_even"><script type="text/javascript">
              googletag.cmd.push(function() { googletag.display('div-gpt-ad-1388181177156-4'); });
              </script></div>
<!-- HOME_CORP_LOGO_6 -->
<div id="div-gpt-ad-1388181177156-5" class="sponsors_google_ads_odd"><script type="text/javascript">
              googletag.cmd.push(function() { googletag.display('div-gpt-ad-1388181177156-5'); });
              </script></div>
</aside>
<div class="grid_12 alpha omega" itemprop="publisher" itemscope="" itemtype="http://schema.org/CollegeOrUniversity">
<h4 class="footer" style="border-top: thin solid #d5c9ba; padding-top: 10px; margin-bottom: 10px;">About <span itemprop="name">MIT OpenCourseWare</span>
</h4>
<p style="color: #999; font-size: 1em; line-height: 1.5em; margin-top: 10px;" itemprop="description">MIT OpenCourseWare is an online publication of materials from over 2,500 MIT courses, freely sharing knowledge with learners and educators around the world. <a href="https://ocw.mit.edu/about/">Learn more »</a></p>
</div>
<div id="foot-copy" class="grid_12 alpha omega" style="border-top: none;">
<a href="http://web.mit.edu"><img src="../../../common/images/logo_mit.png" alt="Massachusetts Institute of Technology logo and name." style="width: 195; height: 44;"></a><a href="https://openlearning.mit.edu/"><img src="https://ocw.mit.edu/images/mitol_logo.png" alt="MIT Open Learning logo and name." style="width: 265; height: 50; vertical-align: top; padding-left:30px;"></a><a href="https://www.oeglobal.org/"><img src="https://ocw.mit.edu/images/oeg_logo.gif" alt="Open Education Consortium logo." style="width: 219px; height: 59px; vertical-align: top; padding-left:20px;"></a><a rel="license" itemprop="useRightsUrl" href="http://creativecommons.org/licenses/by-nc-sa/4.0/"><img src="../../../common/images/cc_by-nc-sa.png" alt="Creative Commons logo with terms BY-NC-SA." style="width: 126px; height: 44px; margin-right: 0; padding-left: 20px;"></a>
<p class="copyright">© 2001–2018<br>
Massachusetts Institute of Technology</p>
<p style="font-size: 0.9em; margin-bottom: 15px;">Your use of the MIT OpenCourseWare site and materials is subject to our <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/" rel="license">Creative Commons License</a> and other <a href="../../../common/terms/index.htm" rel="cc:morePermissions">terms of use</a>.</p>
</div>
</div></div>

</div>





                
			</div> <!-- bottom grid end -->
		</footer><!-- footer bottom end -->


   </body>
 </html>
